The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category

Let $\mathcal{C}$ be a locally small category, so every hom-collection $\mathcal{C}(a,b)$ is a set (Category, object, morphism, domain, codomain, identity, composition, and hom-collection, Small, locally small, and large categories). Since sets and functions form $\mathbf{S}\mathbf{e}\mathbf{t}$ (Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$), the following assignments take values in $\mathbf{S}\mathbf{e}\mathbf{t}$.

For an object $a$, the covariant hom-assignment $\mathcal{C}(a,-)$ sends an object $b$ to $\mathcal{C}(a,b)$ and a morphism $u:b\to c$ to the function

$u_{\ast }:\mathcal{C}(a,b)⟶\mathcal{C}(a,c),f⟼u\circ f\mathrm{.}$

The contravariant hom-assignment $\mathcal{C}(-,a)$ sends $b$ to $\mathcal{C}(b,a)$ and $u:b\to c$ to

$u^{\ast }:\mathcal{C}(c,a)⟶\mathcal{C}(b,a),g⟼g\circ u\mathrm{.}$

Equivalently, the latter is an assignment on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$). Their functor laws are proved in The assignments $\mathcal{C}(a,-)$ and $\mathcal{C}(-,a)$ are functors to $\mathbf{S}\mathbf{e}\mathbf{t}$ ↗.

The two variables combine into the hom-assignment

$\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}⟶\mathbf{S}\mathbf{e}\mathbf{t}\mathrm{.}$

It sends $(a,b)$ to $\mathcal{C}(a,b)$. A morphism $(a,b)\to (a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ in the product category consists of $h:a^{\mathrm{\prime }}\to a$ and $u:b\to b^{\mathrm{\prime }}$ in $\mathcal{C}$ (Product category and its projection functors), and its action is

$\mathcal{C}(h,u):\mathcal{C}(a,b)⟶\mathcal{C}(a^{\mathrm{\prime }},b^{\mathrm{\prime }}),f⟼u\circ f\circ h\mathrm{.}$

That this assignment is a functor, and hence a bifunctor, is proved in The hom-assignment $\mathcal{C}(-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is a bifunctor ↗.