The Stacks project

Remark 59.36.3. More generally, let $\mathcal{C}_1, \mathcal{C}_2$ be sites, and assume they have final objects and fibre products. Let $u: \mathcal{C}_2 \to \mathcal{C}_1$ be a functor satisfying:

1. if $\{ V_ i \to V\}$ is a covering of $\mathcal{C}_2$, then $\{ u(V_ i) \to u(V)\}$ is a covering of $\mathcal{C}_1$ (we say that $u$ is continuous), and

2. $u$ commutes with finite limits (i.e., $u$ is left exact, i.e., $u$ preserves fibre products and final objects).

Then one can define $f_*: \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2)$ by $f_* \mathcal{F}(V) = \mathcal{F}(u(V))$. Moreover, there exists an exact functor $f^{-1}$ which is left adjoint to $f_*$, see Sites, Definition 7.14.1 and Proposition 7.14.7. Warning: It is not enough to require simply that $u$ is continuous and commutes with fibre products in order to get a morphism of topoi.