Given functors $F$ and $G$, does $\mathrm{Res}_F \cong \mathrm{Res}_G$ imply $F \cong G$?

Question

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?

Answer 1

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 $\widehat{\mathbf{D}}$.

Answer 2

If $\mathcal{C}$ and $\mathcal{D}$ are small then $Res_F$ is right adjoint to $F^>:=Lan_{Y_D\circ F} Y_C$ where $Y_C: \mathcal{C}\to \mathcal{C}^>, Y_D: \mathcal{D}\to \mathcal{D}^>$ are the Yoneda immersion's infact:

$\mathcal{D}^>(F^>(P), Q)=\mathcal{D}^>(\varinjlim_{X\in \mathcal{C}\downarrow F}h_{F(X)}, Q)\cong \varprojlim_{X\in \mathcal{C}\downarrow F}Q(F(X))\cong (\varinjlim_{X\in \mathcal{C}}h_X, Q\circ F)\cong (P, Q\circ F)$

or see Popescu, "Theory of Categories" COr.2.7 p.159).

Furthermore $F^>\circ Y_C\cong F$, all this is "natural", then your final assumption is true.