Question. Let $i:\mathcal{C}\hookrightarrow\mathcal{D}$ be a full subcategory. Assume we are given a cocomplete category $\mathcal{A}$. Show that the induced pre-composition functor $i^*:\mathrm{Fun}(\mathcal{D,A})\to \mathrm{Fun}(\mathcal{C,A})$ between the functor categories admits a left adjoint, $i_!:\mathrm{Fun}(\mathcal{C,A})\to \mathrm{Fun}(\mathcal{D,A})$.
We can think of $i^*$ as a 'forgetful' functor. Its left adjoint should perhaps be a functor that 'completes' a functor $F:\mathcal{C}\to \mathcal{A}$ in the 'freest' way possible, which can be obtained by taking
$$i_! (F)(x):= \operatorname{colim}_{a\in i_{/x}} F(a),$$
where $i_{/x}$ is the slice category (its objects are $(a,f)$ where $a\in \mathcal C$ and $f:i(a)\to x$ is a morphism in $\mathcal{D}$; its morphisms $g:(a,f)\to(a',f')$ are $\mathcal C$-morphisms $g:a\to a'$ such that $f'\circ F(g)=f$).
We should verify that for every functors $F:\mathcal{D\to A}$ and $G:\mathcal{C\to A}$, setting a natural transformation $\eta:i_! F\to G$ is "equivalent" to choosing a natural transformation $\epsilon: F\to i^* G$. I am not sure how to proceed.