Co- and Contravariant Hom Functors

For the following let $ \mathcal{C} $ be a locally small category.

The hom-functor (covariant)

Given $ \mathcal{C} $, each object $ A $ determines a functor $$ \mathcal{C}(A,-) : \mathcal{C} \rightarrow Set $$ This functor takes each $ \mathcal{C} $-object $ X $ to the set $ \mathcal{C}(A,X) $, that is the set of arrows from $ A $ to $ X $. Furthermore the functor takes each $ \mathcal{C} $-arrow $ f : X \rightarrow Y $ to the function $$ \mathcal{C}(A,f) : \mathcal{C}(A,X) \rightarrow \mathcal{C}(A,Y) $$ given as $$ \mathcal{C}(A,f)(g : A \rightarrow X) = f \circ g $$ where the right side composition, $ f \circ g $, is in $ \mathcal{C} $. $ \mathcal{C}(A,-) $ is called a hom-functor. This is sketched in Figure 1.

Figure 1: Diagram showing how the covariant hom-functor transforms objects and arrows.

Proof of functor

We can prove that the above is a functor. Fix the object $ A $. Let objects $ X,Y $. All in $ \mathcal{C} $. Let $ F $ be shorthand for $ \mathcal{C}(A,-) $, that is $ F(X) = \mathcal{C}(A,X) $ and for $ f : X \rightarrow Y $ in $ \mathcal{C} $ we have that $ F(f) = \mathcal{C}(A,X) \rightarrow \mathcal{C}(A,Y) $. Now let $ f : X \rightarrow Y $ and $ g : Y \rightarrow Z $ in $ \mathcal{C} $. And we get $$ F(g \circ f) = \mathcal{C}(A,X) \rightarrow \mathcal{C}(A,Z) $$ and $$ F(g) F(f) = (\mathcal{C}(A,Y) \rightarrow \mathcal{C}(A,Z)) (\mathcal{C}(A,X) \rightarrow \mathcal{C}(A,Y)) $$ and hence $$ F(g) F(f) = \mathcal{C}(A,X) \rightarrow \mathcal{C}(A,Z) = F(g \circ f) $$ Furthermore we have that $$ F(id_X) = \mathcal{C}(A,X) \rightarrow \mathcal{C}(A,X) = id_{F(X)} $$

Contravariant hom-functor

The contravariant hom-functor is defined as $$ \mathcal{C}(-,B) : \mathcal{C}^{op} \rightarrow Set $$ that takes each object $ X $ to the set $ \mathcal{C}(X,B) $. Each arrow $ h : X \rightarrow Y $ is taken to the function $$ \mathcal{C}(h,B) : \mathcal{C}(X,B) \rightarrow \mathcal{C}(Y,B) $$ given as $$ \mathcal{C}(h,B)(g : X \rightarrow B) = g \circ h $$ As shown in Figure 2. Note that since $ h \in \mathcal{C}^{op} $, we have the reverse version as $ h : Y \rightarrow X \in \mathcal{C} $. Hence we have that $ g \circ h \in \mathcal{C}(Y,B) $.

Natural relation between covariant and contravariant

The two above are related in a natural manner. That is for every $ \mathcal{C} $-arrows $ f : B \rightarrow B' $ and $ h : A' \rightarrow A $ the following diagram commutes

Diagram of covariant and contravariant hum-functors resulting in a natural transformation.

Bivariant hom-functor

The bivariant hom-functor is for objects defined as $$ \mathcal{C}(-,-) : \mathcal{C}^{op} \times \mathcal{C} \rightarrow Set $$ This is just a composite of the two above functors.