I have the following problem. Let $\mathcal{C}_0$, $\mathcal{C}$ be small categories and $\mathcal{D}$, $\mathcal{E}$ be locally small categories. Let $Q\colon \mathcal{C}_0\to \mathcal{C}$ be a functor and let $$ Q_!\colon [\mathcal{C}_0,\mathcal{D}]\rightleftarrows [\mathcal{C}_0,\mathcal{D}]\colon Q^\ast $$ be a corresponding adjunction between functor categories, where the right adjoint is the pullback and the left adjoint is the left Kan extension along $Q$. Similarly, we have the adjunction $$ Q_!\colon [\mathcal{C}_0,\mathcal{E}]\rightleftarrows [\mathcal{C}_0,\mathcal{E}]\colon Q^\ast. $$
Further on, let $F\colon \mathcal{D}\to \mathcal{E}$ be a functor. Now the question is: if I know that the following diagram is commutative (note the direction of arrows), where $F_\ast$ denotes composition with $F$ $\require{AMScd}$ \begin{CD} [\mathcal{C}_0,\mathcal{D}] @<{Q^\ast}<<[\mathcal{C},\mathcal{D}]\\ @V F_\ast V V @VV F_\ast V\\ [\mathcal{C}_0,\mathcal{E}] @<{Q^\ast}<< [\mathcal{C},\mathcal{E}] \end{CD}
under what conditions I can derive that the following diagram is also commutative? $\require{AMScd}$ \begin{CD} [\mathcal{C}_0,\mathcal{D}] @>Q_!>> [\mathcal{C},\mathcal{D}]\\ @V F_\ast V V @VV F_\ast V\\ [\mathcal{C}_0,\mathcal{E}] @>>Q_!> [\mathcal{C},\mathcal{E}] \end{CD}
A remark: specific example which I have in mind is that $D$ is the category of topological spaces, $E$ is the category of chain complexes of abelian groups and $F$ is the functor of singular chains. If the above question is too general, I would be happy with an idea/reference how to prove only this case.
This question is a follow-up to this one: Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free.
