The Unapologetic Mathematician

Functor Categories

Let’s consider two categories enriched over a monoidal category $\mathcal{V}$$\mathcal{C}$ and $\mathcal{D}$ — and assume that $\mathcal{C}$ is equivalent to a small category. We’ll build a $\mathcal{V}$-category $\mathcal{D}^\mathcal{C}$ of functors between them.

Of course the objects will be functors $F:\mathcal{C}\rightarrow\mathcal{D}$. Now for functors $F$ and $G$ we need a $\mathcal{V}$-object of natural transformations between them. For this, we will use an end:
$\hom_{\mathcal{D}^\mathcal{C}}(F,G)=\int_{A\in\mathcal{C}}\hom_\mathcal{D}(F(C),G(C))$
This end is sure to exist because of the smallness assumption on $\mathcal{C}$. Its counit will be written $E_C=E_{C,F,G}:\hom_{\mathcal{D}^\mathcal{C}}(F,G)\rightarrow\hom_\mathcal{D}(F(C),G(C))$.

An “element” of this $\mathcal{V}$ object is an arrow $\eta:\mathbf{1}\rightarrow\hom_{\mathcal{D}^\mathcal{C}}(F,G)$. Such arrows correspond uniquely to $\mathcal{V}$-natural families of arrows $\eta_C=E_C\circ\eta:\mathbf{1}\rightarrow\hom_\mathcal{D}(F(C),G(C))$, which we know is the same as a $\mathcal{V}$-natural transformation from $F$ to $G$. We also see that at this level of elements, the counit $E_C$ takes a $\mathcal{V}$-natural transformation and “evaluates” it at the object $C$.

Now we need to define composition morphisms for these hom-objects. This composition will be inherited from the target category $\mathcal{B}$. Basically, the idea is that at each object a natural transformation gives a component morphism in the target category, and we compose transformations by composing their components. Of course, we have to finesse this a bit because we don’t have sets and elements anymore.

So how do we get a component morphism? We use the counit map $E_C$! We have the arrow
$E_C\otimes E_C:\hom_{\mathcal{D}^\mathcal{C}}(G,H)\otimes\hom_{\mathcal{D}^\mathcal{C}}(F,G)\rightarrow\hom_\mathcal{D}(G(C),H(C))\otimes\hom_\mathcal{D}(F(C),G(C))$
which we can then hit with the composition $\circ:\hom_\mathcal{D}(G(C),H(C))\otimes\hom_\mathcal{D}(F(C),G(C))\rightarrow\hom_\mathcal{D}(F(C),H(C))$. This is a $\mathcal{V}$-natural family indexed by $C\in\mathcal{C}$, so by the universal property of the end we have a unique arrow
$\circ:\hom_{\mathcal{D}^\mathcal{C}}(G,H)\otimes\hom_{\mathcal{D}^\mathcal{C}}(F,G)\rightarrow\hom_{\mathcal{D}^\mathcal{C}}(F,H)$

Similarly we get the arrow picking out identity morphism on $T$ as the unique one satisfying $E_C\circ i_T=i_{T(A)}$

So $\mathcal{D}^\mathcal{C}$ is a $\mathcal{V}$-category whose underlying ordinary category is that of $\mathcal{V}$-functors and $\mathcal{V}$-natural transformations between the $\mathcal{V}$-categories $\mathcal{C}$ and $\mathcal{D}$. That is, it’s the hom-category $\hom_{\mathcal{V}\mathbf{-Cat}}(\mathcal{C},\mathcal{D})$ in the 2-category of $\mathcal{V}$-categories.