Let $C$ be a small category. Let $D,E$ be cocomplete categories. Let us denote by $\hom$ (resp. $\hom_c$) the category of (cocontinuous) functors. Then there is an equivalence of categories $$\hom_c\bigl(\hom(C,D),E\bigr) \simeq \hom_c\bigl(D,\hom(C^{\mathrm{op}},E)\bigr).$$ In other words, $\hom(C,-) : \mathsf{Cat}_c \to \mathsf{Cat}_c$ is $2$-left adjoint to $\hom(C^{\mathrm{op}},-) : \mathsf{Cat}_c \to \mathsf{Cat}_c$.
The equivalence maps a cocontinuous functor $H : D \to \hom(C^{\mathrm{op}},E)$ to $\tilde{H} : \hom(C,D) \to E$, $$\tilde{H}(F) = \int^{c \in C} H(F(c))(c).$$ Conversely, a cocontinuous functor $\tilde{H} : \hom(C,D) \to E$ is mapped to $H : D \to \hom(C^{\mathrm{op}},E)$, $$H(d)(c) = \tilde{H}(\hom(c,-) \times d).$$
Question. Does someone have a reference for this adjunction?
For $D=\mathsf{Set}$ it is well-known, then we get the usual adjunction (after exchanging $C$ with its dual) $\hom_c(\widehat{C},E) \simeq \hom(C,E)$.