Assume $F, G : \mathbf C \to \mathbf D$ be functors. Denote by $\widehat{\mathbf C} = \mathrm{Fun}(\mathbf{C}^{\mathrm{op}}, \mathbf{Set})$ the category of presheaves of sets on $\mathbf C$. Then, $F$ and $G$ induce "restriction" functors, obtained by composition with $F$ and $G$:
\begin{align\*}
\mathrm{Res}\_F, \mathrm{Res}\_G : \widehat{\mathbf D} \to \widehat{\mathbf C}.
\end{align\*}
If $F \cong G$ then clearly $\mathrm{Res}\_F \cong \mathrm{Res}\_G$. I believe that the converse is false; how can I find a counterexample? Or perhaps it is true under certain assumptions?



Given functors $F$ and $G$, does $\mathrm{Res}_F \cong \mathrm{Res}_G$ imply $F \cong G$?
You can recover $F$ from $\mathrm{Res}\_F$; this is an exercise in using the Yoneda lemma. Let $H\_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \mathbf{C}$ and $B\in\mathbf{D}$ we have $$\operatorname{Hom}\_{\mathbf D}(FA,B)=H\_B(FA)=\mathrm{Res}\_FH\_B(A).$$
By Yoneda, the left hand side is enough to recover the functor $F$.


We can also use Yoneda again to rewrite the left-hand side as $\operatorname{Hom}\_{\widehat{\mathbf D}}(H\_{FA},H\_B)$ and the right-hand side as $\operatorname{Hom}\_{\widehat{\mathbf C}}(H\_A,\mathrm{Res}\_FH\_B)$. This suggests that $\mathrm{Res}\_F$ has a left adjoint which is given by $H\_A\mapsto H\_{FA}$ on representable presheaves. When our categories are small, this is true: $\mathrm{Res}\_F$ has a left adjoint which is the left Kan extension of the composition $\mathbf{C}\to\mathbf{D}\to\widehat{\mathbf{D}}$ along $\mathbf{C}\to\widehat{\mathbf{C}}$. This adjoint coincides with $F$ on representable presheaves and in general is computed by writing any presheaf as a colimit of representable presheaves, applying $F$ to each of the representing objects, and taking the corresponding colimit in $\widehat{\mathbf{D}}$. 


