The Stacks project

Remark 7.44.4. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a continuous functor which gives rise to a morphism of sites $\mathcal{C} \to \mathcal{D}$. Note that even in the case of abelian groups we have not defined a pullback functor for presheaves of abelian groups. Since all colimits are representable in the category of abelian groups, we certainly may define a functor $u_ p^{ab}$ on abelian presheaves by the same colimits as we have used to define $u_ p$ on presheaves of sets. It will also be the case that $u_ p^{ab}$ is adjoint to $u^ p$ on the categories of abelian presheaves. However, it will not always be the case that $u_ p^{ab}$ agrees with $u_ p$ on the underlying presheaves of sets.