contravariant hom functor

Simion Breaz, Corresponding Author. If you have an Encoder[Long] and a function Int => Long, you can create an Encoder[Int] using contramap.Internally, the contramapped encoder calls the function to . For the contravariant Hom functor M → Hom(M,X), with X = C and F : C → X the identity, the exactness of the Hom sequence gives 0 = F g f = g f Thus, Imf ⊂ kerg. Covariance and Contravariance in Scala - Work Life by ... For an example of a contravariant functor, consider the functor Spec: Ring !Sch. C-mod: Let us remark that the functor is a right exact covariant functor, while the functor 1 is a left exact covariant functor. A the functor HomC( ;A) (recall that this is a contravariant functor). Also we may state a similar result for the functor Hom A ( X, −). We then define a contravariant functor F The inverse Nakayama functor 1 is de ned to be the composition Hom Cop( ;A) D: C-fdmod ! As usual, we write [sup. Action as a contravariant functor # Wrapping an Action<T> in a DelegatingCommandHandler isn't necessary in order to form the contravariant functor. derive functor · Wiki · Glasgow Haskell Compiler / GHC ... Active 9 years, 10 months ago. One that performs an operation or a function. Example: (Hom functor) Recall that if A;Bare objects of a category C, then Hom C (A;B) denotes the set of all morphisms from Ato B. hom-functor in nLab The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. I only used the ICommandHandler interface as an object-oriented-friendly introduction to the example . Example #4: contravariant functors. contravariant: Contravariant functors - Haskell Given a ring A, SpecAis the set of prime ideals of A, viewed as a scheme. A functor from a category to itself is called an endofunctor. DIRECT PRODUCTS AND THE CONTRA V ARIANT HOM-FUNCTOR SIMION BREAZ Abstract. PDF Lecture 2: Functors - UMinho Covariant Functor A functor is called covariant if it preserves the directions of arrows, i.e., every arrow is mapped to an arrow . 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits . Let's implement a co-vector field using a contravariant functor. Exact functor - Academic Kids Where the (contravariant) Functor is all functions with a common result - type G a = forall r. a -> r here the Contravariant instance would be cmap ψ φ = φ . Direct products and the contravariant hom-functor ... A contravariant functor G from C to Set is the same thing as a functor G : C op → Set and is commonly called a presheaf. Then a map of rings A!Bgives a map of schemes SpecB!SpecA. contravariant. (The phrase \set of all:::" must be taken with a grain of logical salt to avoid the well known paradoxes of set theory. A bifunctor (also known as a binary functor) is a functor whose domain is a product category. Twan van Laarhoven first proposed this feature in 2007, and opened a related GHC ticket in 2009. We have the following basic but crucial lemma. Theorem: Let Cbe a category. seek a scheme that coarsely represents the functor. A lot of people think of an F[A] as something that 'contains' or can produce an A, but for a contravariant functor, it's something that can consume an A.. A common concrete example is an Encoder[A] that can convert an A to json. The bottom map is induced by a similar composite with G in place of F. The dual notion gives an enriched hom of V-functors G : C !D and H : C !D From Wikipedia, the free encyclopedia. An imbedding functor (cf. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Let X be a projective scheme over a noetherian base scheme S, and let F be a coherent sheaf on X. What is a functor in programming? In this post we're going to be exploring the idea of enhancing normal data types with different types of functor structures step-by-step, by starting with a simple useful structure and enhancing it piece by piece . a;bis the arrow of V given because F is a V-functor and ev is the evaluation map, the counit of the adjunction on V of the monoidal product with the internal-hom. Please check this. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. At the same time, we can completely forget, that F[_] is a functor - we only see that Yoneda[F, ?] What we have called \functor" so far is also called a covariant functor. Examples 2.3. But we should warn the reader that the functor 1 is, in general, neither the inverse nor a quasi-inverse of . The theorem (above) characterizing natural transformations from a representable functor to an arbitrary functor is commonly called the Yoneda lemma. Direct products and the contravariant hom-functor @article{Breaz2012DirectPA, title={Direct products and the contravariant hom-functor}, author={S. Breaz}, journal={Bulletin of The London Mathematical Society}, year={2012}, volume={44}, pages={136-138} } Hom is a bifunctor Similarly if B is a ﬁxed module then Hom( ;B) is a functor from R-modules to sets, but this functor iscontragredient! Between categories A contravariant functorFFfrom a categoryCCto a category DDis simply a functorfrom the opposite categoryCopC^opto DD. There is a compatibility between the two functors Hom(A; ) and Hom( ;B). Haskell 98 contravariant functors. To show that the contravariant $\text{Hom}$ functor is left-exact. . More particularly, the functor Hom^,: 0Z°v x0l H» gnu extends in a natural way to a functor ^op x M-> 01. (ZFC) that if G is a (right) R-module such that the groups Hom R . A contravariant functor is like a functorbut it reverses the directions of the morphisms. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(-,A) for some object A of C. Universal elements. Hom(X; ) is left exact Everything later will reduce to the straightforward left-exactness of Hom(X; ). There is a functor F: Vectk → Vectk F: V e c t k → V e c t k where Vectk V e c t k consists of all vector spaces over a field k k with linear transformations as morphisms. This F F sends a vector space V V to hom(V,k) hom ( V, k), the space of linear functions V → k V → k. φ`. The functor F A is exact if and only if A is projective. Str. Imbedding of categories) from a category $\mathcal {C}$ into the category $\hat {\mathcal {C}}$ of contravariant functors defined on $\mathcal {C}$ and taking values in the category of sets $\mathsf {Ens}$. (The standard nomenclature is X o p ). Welcome to Part 2 of the "Enhancing Functor Structures" series! The functor G A (X) = Hom A (X,A) is a contravariant left-exact functor; it is exact if and only if A is injective. There is no real conceptual difference between "contravariant" and "covariant" because of the duality of abelian categories; a contravariant functor is just a functor on the dual category. an object. Contravariant functors often occur when there is some kind of duality in the picture. arXiv:1105.4399v1 [math.RA] 23 May 2011 DIRECT PRODUCTS AND THE CONTRAVARIANT HOM-FUNCTOR SIMIONBREAZ Abstract. adshelp[at]cfa.harvard.edu The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A For example, the Hom functor is of the type Cop × C → Set. The functor Hom Let Abe a ring (not necessarily commutative).Consider the collection of all left A-modules Mand all module homomorphisms f: M!Nof left A-modules. Grammar See function word. For instance in the one-object case, obtained from a ring R= End(), a functor from Ato Ab is determined by the image of , an abelian group - let us denote it Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange It is covariant in B but contravariant in A. This is a composable pair, and we may form the composition f g. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. Here we are taking a base structure describing a data type schema and enhancing it step-by-step with new functory capabilities: first, covariant capabilities (to generate parsers), then contravariant capabilities (to generate serializers)…who knows what might be in store next? For the contravariant Hom functor M → Hom(M,X), with X = C and F : C → X the identity, the exactness of the Hom sequence gives 0 = F g f = g f Thus, Imf ⊂ kerg. GHC 6.12.1 introduces an extension to the deriving mechanism allowing for automatic derivation of Functor, Foldable, and Traversable instances using the DeriveFunctor, DeriveFoldable, and DeriveTraversable extensions, respectively. As far as the functorality of "->" goes: from the fact that (->) is a profunctor, it follows that (A->) is a covariant functor and that (->B) is a contravariant functor, for any choice of A and B. Let Cbe any category. (i) via a non-reversed arrow in the domain category ($\mathcal{C}$) plus a contravariant hom-functor; (ii) via an already-reversed arrow in the domain category plus a covariant hom-functor, where the arrow reversing is done by the op-trick which changes the domain category from $\mathcal{C}$ to $\mathcal{C}^{op}$. f. where g is in Hom (B', B), rather than Hom (A, B). Example 1.13 (Spec, a contravariant functor). Support for deriving Functor, Foldable, and Traversable instances. Examples Diagram: For categories C and J, a diagram of type J in C is a covariant functor . if F(f) is an isomorphism so is f Exercise 17 φ :: a -> b and ψ :: b -> c. . The definition (1) describes a linear map between a vector V over a field X to the scalar product V*: V => T. A morphism on the category V* consists of a morphism of V => T or V => _ where V is a vector field and T or _ is a scalar function value. For example, the algebra \Gamma(X, \mathcal{O}_X) of regular functions on a projective variety X over a field is rather boring; it's ju. And so the hom functor on vector spaces is a contravariant functor, while all of the other functors we've defined in this post are covariant. The other way (used in Cats and Scalaz documentation) is that it is partially applied map method of a Functor type class, there the f is passed only after you finished composing functions. A contravariant functor G from C to Set is the same thing as a functor G : C op → Set and is commonly called a presheaf. Similarly, we denote the functor HomC(A; ) by hA. Definition: A functor is called covariant if it preserves the order of morphism composition, so that . On the other hand, with X = B/Imf and F : B → X the quotient map, by exactness of the Hom sequence there is F0: C → X such that F0 g = F. Thus, the kernel of g cannot be . There are many constructions in mathematics which would be functors but for the fact that they "turn morphisms around" and "reverse composition". I've been trying to understand some of the abstract concepts in Haskell by translating them to Python, but I can't seem to figure out how to write a class that satisfies the axioms defining contravariant functor. Imbedding of categories) from a category $\mathcal {C}$ into the category $\hat {\mathcal {C}}$ of contravariant functors defined on $\mathcal {C}$ and taking values in the category of sets $\mathsf {Ens}$. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. lar, a functor F: C! Hom D(F(X);F(Y)) is injective (respectively, surjective). The purpose of the article was mainly to give you a sense of what a contravariant functor can do. 2. So, for example, a bifunctor is a multifunctor with n = 2 . If A is an abelian category, then Hom(A,−) is a covariant, additive, kernel preserving functor A −→ Ab and Hom(−,B) is a contravariant, additive functor which maps cokernels to kernels. Lemma 2.4. On the other hand, with X = B/Imf and F : B → X the quotient map, by exactness of the Hom sequence there is F0: C → X such that F0 g = F. Thus, the kernel of g cannot be . The question is whether the right, (perhaps left . More precisely, since the hom-functor is contravariant in the first variable, when we fix the target object, it maps colimits in the first variable to limits. What is a functor category? If k is a field and V is a vector space over k, we write V* = Hom k (V,k). In particular we can prove that for the contravariant functor F = Hom A (−, X ): A → Ab we have R nωF = ωExt nA( −, X) (where ω is a projective p.c.). n. 1. Hom D(F(X);F(Y)) is injective (respectively, surjective). These Hom functors need not be exact, but as we shall see the modules Ufor which they are exact play a very important role in our study. This is an artifact of the way in which one must compose the morphisms. Viewed 2k times 4 4 $\begingroup$ Hi, I apologise if this is the wrong place for this question but i need to ask it somewhere. EXAMPLES 8.2.2. a. A left A-module is a functor from Ato the category, Ab, of abelian groups. Active 3 years, 8 months ago. Please feel free to contact me through github or on the #haskell IRC channel on irc.freenode.net. We prove in ZFC that if Gis a (right) R-module such that the groups Hom R(Q i∈IG , G) and Q i∈IHomR(G , G) are naturally. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. Now suppose that f: A → B, g: B → C in X. For example, Ext 0 R (A, B) is the kernel of the map Hom R (A, I 0) → Hom R (A, I 1), which is isomorphic to Hom R (A, B). If T is a contravariant cohomological δ -functor with T d + 1 = 0, then T d is an example of a contravariant right-exact functor. G(f) is a homomorphism from G(Y) to G(X) instead of the other way around. I think the following class captures the concept of a basic functor on 'non-iterable' types. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(-,A) for some object A of C. Direct products and the contravariant hom-functor Simion Breaz. It can be seen as a functor in two arguments. So Hom (-, B) acts as a contravariant functor. hom : C^ {op} \times C \to Set. Grothendieck functor. Show that full and faithful functors reﬂect and create isomorphisms, i.e. Given a contravariant functor F from schemes over Sto sets, we say that a scheme X(F) over Scoarsely represents the functor Fif there is a natural transformation of functors: F!Hom S(;X(F)) such that (1)( spec(k)) : F(spec(k)) !Hom S(spec(k);X(F)) is a bijection [disjunction]] for the contravariant functor Hom(-, R). The statement of the Yoneda Lemma (in contravariant form) is the following. Hom (f, B) (g) = g . De nition 1.5. Hilbert-Samuel polynomials for the contravariant extension functor - Volume 198 Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. on 2, and the powerset Pis seen to be a contravariant functor to the category of Boolean algebras, PBA: Setsop!BA: As was the case for the covariant representable functor Hom Grp(F(1); ) and the forgetful functor Ufrom groups to sets, here the contravariant functors Hom Sets( ;2) and PBA from sets to Boolean algebras can also be seen to be We give some examples of functors. (ZFC) that if G is a (right) R-module such that the groups Hom R . Hom functors An arrow in (locally small) induces a function: f: A → B C for any object .X f −: hom(X,A) → hom(X,B) - This gives a covariant hom-functor : A multifunctor is a generalization of the functor concept to n variables. Example 1.14 (Hom). b) If it reverses the order, we call it contravariant. Contact Information. Throughout this section A will be an abelian category with enough injectives. We prove in ZFC that if G is a (right) R-module such that the bodo@math.ubbcluj.ro; Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. Grothendieck functor. Show that full and faithful functors reﬂect isomorphisms, i.e. A contravariant functor G from C to Set is the same thing as a functor G : C op → Set and is therefore representable just when it is naturally isomorphic to the contravariant hom-functor Hom(-,A) for some object A of C. Universal elements More particularly, the functor Hom^,: 0Z°v x0l H» gnu extends in a natural way to a functor ^op x M-> 01. Contributions and bug reports are welcome! The right and left derived functors of contravariant functors can be defined by the duality. S. MacLane [a1] traces their first appearance to work of J.-P. Serre in algebraic topology, around 1953. But what the hell does this mean. This yields an exact functor from the category of k-vector spaces to itself. Dually, a left module RQis injective in case the contravariant duality functor Hom R . Dis faithful (respectively, full) if the map Hom C(X;Y) ! Formally, a bifunctor is a functor whose domain is a product category. To summarize, Hom(A;B) is a functor in both A and B. covariant functor. The contravariant hom functor, C( ;X) : Cop!Set sends any Y 2obCopto the set of morphisms from Y to X. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania . SEE ALSO: Contravariant Functor , Forgetful Functor , Functor , Hom , Tensor Product Functor Let Rbe a ring. The most important examples of left exact functors are the Hom functors: if A is an abelian category and "A" is an object of A, then "F" "A" ("X") = Hom A ("A","X") defines a covariant left-exact functor from A to the category Ab of abelian group s. An embedding is a faithful functor which is, additionally, injective on morphisms. Suppose that f : A ! The Hom-Functor The above examples are the reflection of a more general statement that the mapping that takes a pair of objects a and b and assigns to it the set of morphisms between them, the hom-set C(a, b) , is a functor. As always the instance for (covariant) Functor is just fmap ψ φ = ψ . lar, a functor F: C! The right derived functors of Hom(A,−) are called the Ext . Is there a right adjoint to the contravariant functor Hom(-,B) in the category of Sets. Please do check out Part 1 if you haven't . That is, a short exact sequence 0 /A i /B q /C /0 gives an exact sequence DOI: 10.1112/blms/bdr083 Corpus ID: 119575078. [1.0.1] Claim: The functor Hom(X; ) is left exact. Ask Question Asked 9 years, 10 months ago. Let Vectk denote the category of small vector spaces over some ﬁxed ﬁel d k. The map which takes each vector space V to its dual V∗ =Hom(V,k)and each linear map to its dual is a contravariant functor from Vectk to Vectk. Direct products and the contravariant hom-functor. For example, the Hom functor is of the type C op × C → Set. The functor Hom (-, B) is also called the functor of points of the object B . For any g: Y !Y0, C( ;X) gives us a map C(Y0;X) !C(Y;X) by sending f 2C(Y0 . Faculty of Mathematics and Computer Science. convention throughout is that \functor" means additive functor. A left module RP is projective in case the covariant evaluation functor Hom R(P;¡):RMod ¡!Ab is exact. Given an object X of C, we can consider the (contravariant) functor of points associated to X: hX: Cop!Set (1) T 7!Hom C(T, X) (2) Note that h_ deﬁnes a covariant functor C!Fun(Cop,Set): if a : X !Y is a morphism, then ha: Hom C( , X) !Hom C( ,Y) is given by composition with a. We denote the functor R h> Horn ^(B, R) by B*, and such a functor from 01 to 0k we call representable. Fix X2obC. In functional programming, a functor is a design pattern inspired by the definition from category theory, that allows for a generic type to apply a function inside without changing the structure of the generic type. Contravariant functor synonyms, Contravariant functor pronunciation, Contravariant functor translation, English dictionary definition of Contravariant functor. 400084 Cluj-Napoca. hom: C op × C → Set. An alternative definition uses the functor G(A)=Hom R (A, B), for a fixed R-module B. Here are examples of a contravariant functor and a covariant func-tor. C. C with its opposite category to the category Set of sets, which sends. As with a covariant functor, a contravariant functor F : C → D is again a map on objects, but with maps between sets of morphisms of the form F: Hom C (A, B) → Hom D (F (B), F (A)) that satisfies F (id A) = id F (A) and F (g f) = F (f) F (g). In category theory, a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors . Answer (1 of 2): One way to view the role of sheaves (and presheaves) in geometry is that they capture local and global information about structures on a space. A contravariant functor G is similar function which reverses the direction of arrows, i.e. This is a contravariant functor, which can be viewed as a left exact functor from the opposite category (R-Mod) op to Ab. Ask Question Asked 3 years, 8 months ago. We will omit the C when the category is clear from context. Simion Breaz Babeş-Bolyai University. A style of Haskell programming that I've been pretty excited about with over the past two years or so is something that I can maybe call a "functor structure" design pattern. An imbedding functor (cf. For any coherent sheaf E on X, consider the set-valued contravariant functor Hom (E,F) on S-schemes, defined by Hom (E,F)(T) = Hom(ET,FT) where ET and FT are the pull-backs of E and F to XT = X ×S T. Let C be any category and Aany object of C. Then Hom(A; ) de nes a covariant functor and Hom( ;A) de nes a contravariant functor from C . That is to say, if B is a biring and ^ a ring, then the co-ring structure of B induces a ring structure on the set Hom^-B, R). is a functor and use it as such. 6 Which makes Hom (A, B) contravariant in A, and covariant in B — just like Function1! Mihail Kogălniceanu 1. Covariance and contravariance. Dis faithful (respectively, full) if the map Hom C(X;Y) ! A bifunctor (also known as a binary functor) is a functor in two arguments. Representable functors occur in many branches of mathematics besides algebraic geometry. Fix a commutative ring R and an R-module M. Then Hom( ;M) is a . This is actually a more general result, since it applies in any category, and not just in the category of types with subtyping. We denote the functor R h> Horn ^(B, R) by B*, and such a functor from 01 to 0k we call representable. Deﬁnition 1. d o m ( f ′) = c o d ( f) c o d ( f ′) = d o m ( f) The opposite X ′ of a category X has the same objects as X, and its system of morphisms consists of all the twins of the morphisms in X. 7. (It also maps limits to limits in the second variable). C. C a locally small category, its hom-functor is the functor. Viewed 2k times 7 1 $\begingroup$ (We're working over $\mathcal{Ab}$) Essentially, I need to show that if the following sequence is exact:$$0\rightarrow A\stackrel{f}{\longrightarrow}B\stackrel{g . EXAMPLE 4.2.4. In either case a functor is homotopy invariant if it takes isomorphic values on homotopy equivalent spaces and sends homotopic maps to the same homomorphism. CHAPTER VI HOM AND TENSOR 1. from the product category of the category. Since a coend is a colimit, and an end is a limit, continuity leads to the following identity: (Between groupoids, contravariant functors are essentially the same as functors.) Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. ψ. Hom(X; ) is left exact Adjoints and naturality A small Yoneda lemma Half-exactness of adjoints 1. These two families arise from the slice categories (aka, comma categories) and are known as the co-/contravariant hom-functors. If F is an arbitrary contravariant functor Cop!Set, then one has HomFun(h A;F) ˘=F(A): For. That is to say, if B is a biring and ^ a ring, then the co-ring structure of B induces a ring structure on the set Hom^-B, R). if F(f) is an isomor- An embedding is a faithful functor which is, additionally, injective on morphisms. Derived functors of Hom ( -, R ) straightforward left-exactness of Hom ( a ; ) and Hom ;. 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Feel free to contact me through github or on the # haskell IRC channel on irc.freenode.net, sends... With enough injectives Asked 9 years, 8 months ago ) to G ( X Y. Which sends van Laarhoven first proposed this feature in 2007, and in... Question Asked 3 years, 10 months ago covariant func-tor ) by hA additionally! Map Hom C ( X ; Y ) to G ( a, the! Section a will be an abelian category with enough injectives - hom-functor /a! Faithful functor which is, in general, neither the inverse nor a of! Surjective ) RQis injective in case the contravariant hom-functor < /a > Let Cbe any category //www.quora.com/Why-are-presheaves-contravariant-functors share=1. First appearance to work of J.-P. Serre in algebraic topology, around.! Contravariant functorFFfrom a categoryCCto a category DDis simply a functorfrom the opposite categoryCopC^opto DD ). A fixed R-module B B & # x27 ;, B ) X o p ) similar result for contravariant! 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G ( F ( Y ) to G ( Y ) ) is left exact Everything later reduce... With n = 2 ) that if G is a natural example ; it is contravariant in,... Instead of the functor HomC ( a ; ) is a ( X ) ; F ( )! Of k-vector spaces to itself type C op × C → Set yields an exact functor from Ato category. Than Hom ( a, B ) acts as a functor in both a and B contravariant... The standard nomenclature is X o p ) enough injectives the Ext to in! Why are presheaves contravariant functors [ a1 ] traces their first appearance to work of J.-P. Serre in topology! With n = 2 left-exactness of Hom ( X ; ) and are known as the hom-functors. C with its opposite category to the straightforward left-exactness of Hom ( a, )! Map Hom C ( X ; Y ) to G ( Y ) ) is the following 1.0.1! ( it also maps limits to contravariant hom functor in the second variable ) > Hom and TENSOR 1 ) in... Https: //runtheyear2016.com/2019/09/22/what-is-a-functor-example/ '' > PDF < /span > Hom and TENSOR 1, Faculty of Mathematics Computer! This section a will be an abelian category with enough injectives 1 if haven...: C for an example of a basic functor on & # x27 ; types it. Form ) is a covariant func-tor the functor 1 is, in,! ; B ) is a multifunctor is a ( B & # x27 ;, B contravariant hom functor contravariant in argument!: //www.reddit.com/r/scala/comments/q4kzju/examples_of_contravariant_functors/ '' > hom-functor in nLab < /a > CHAPTER VI Hom and TENSOR...., full ) if the map Hom C ( X ; Y contravariant hom functor injective ( respectively, full ) the! A will be an abelian category with enough injectives left A-module is a contravariant hom functor functor which,... Embedding is a faithful functor which is, additionally, injective on morphisms TENSOR 1, Hom -... Categories a contravariant functor and a covariant functor times C & # 92 ; to Set → C in.... Where G is a functor in both a and B the opposite categoryCopC^opto DD clear from context 1.0.1! C in X around 1953 is in Hom ( contravariant hom functor ) ; (! > Let Cbe any category: a → B, G: →.