Let $Q$ be a quiver and ${C}_1, C_2$ be two (small) categories. Let $F:C_1\rightarrow C_2$ and $G:C_2\rightarrow C_1$ be two (covariant) functors.
- Is it true that if $F$ and $G$ induce functors $$ F_Q: \text{Rep}_{C_1}(Q)\rightarrow\text{Rep}_{C_2}(Q) \quad \text{ and } \quad G_Q: \text{Rep}_{C_2}(Q)\rightarrow\text{Rep}_{C_1}(Q)? $$ Here $\text{Rep}_C(Q)$ means the category of representations of $Q$ over the category $C$.
- If so, is it true that if $F$ is a left adjoint of $G$, then $F_Q$ is a left adjoint of $G_Q$?
A demonstration is welcome, but if it is true I just need a reference as, for instance, a book or a paper.
Given a quiver $Q$, there is an associated small category $\mathcal P$ (which is just the "free category" on the quiver -- you add an identity morphism for each vertex, and compositions of directed edges etc.). The category $\mathrm{Rep}_C(Q)$ is then the functor category $C^{\mathcal P}$.
Phrased this way your questions become fairly standard: your first question is just a matter of understanding functor categories --- see e.g. chapter 2 of Riehl's book -- while the second question is a consequence of basic properties of adjunctions, see e.g. chapter 4 of Riehl's book.