Let $\textbf{Cat}$ denote the category of small categories and functors between them. Fix $\mathcal{C}\in\textbf{Cat}$. I want to construct a functor
$$[\mathcal{C},-]:\textbf{Cat}\rightarrow\textbf{Cat}$$
analogous to the hom-functors.
Obviously, $[\mathcal{C},-](\mathcal{D}):=[\mathcal{C},\mathcal{D}]$ for all $\mathcal{D}\in\textbf{Cat}$.
Let $F:\mathcal{D}\rightarrow\mathcal{D}'$ be a functor between two small categories. Then $[\mathcal{C},-](F):=[\mathcal{C},F]$ has to be a functor from $[\mathcal{C},\mathcal{D}]$ to $[\mathcal{C},\mathcal{D'}]$. For a functor $G:\mathcal{C}\rightarrow\mathcal{D}$, we can set $[\mathcal{C},F](G):=F\circ G$. Now, let $\alpha:G\Rightarrow H$ be a natural transformation between $G,H:\mathcal{C}\rightarrow\mathcal{D}$. Should I define $[\mathcal{C},F](\alpha):=F*\alpha$? ($*$ denotes the Godement product)
Edit:
Note that $F*\alpha$ actually denotes $1_F*\alpha$, where $1_F$ is the identity natural transformation on $F$.
Best Answer
Well, if $\mathcal C=*$ is the terminal category, then you presumably want $[*,-]$ to be naturally isomorphic to the identity functor via evaluation at the only object of $*$. In this case a morphism $\alpha$ in $[*,\mathcal D]$ is identified with a morphism in $\mathcal D$, and the Godement product $F*\alpha$ is identified with $F(\alpha)$, so this looks good.
To conclude for a general $\mathcal C$, you just have to insist that you want $[\mathcal C,F]$ to be natural in $\mathcal C$. Then for any object of $\mathcal C$, viewed as a functor $c:*\to \mathcal C$, and any morphism $\alpha$ in $[\mathcal C,\mathcal D]$, we find $[\mathcal C,F](\alpha)_{c}=F(\alpha_c)$, as desired. The relevant naturality square here, to be clear, is $$\require{AMScd} \begin{CD} [\mathcal C,\mathcal D] @>{[\mathcal C,F]}>> [\mathcal C,\mathcal D'];\\ @V{[c,\mathcal D]}VV @V{[c,\mathcal D']}VV \\ [*,\mathcal D]@>[*,F]>> [*,\mathcal D']; \end{CD}$$