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?