The Stacks project

Definition 7.15.1 (Topoi). A topos is the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of sheaves on a site $\mathcal{C}$.

1. Let $\mathcal{C}$, $\mathcal{D}$ be sites. A morphism of topoi $f$ from $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is given by a pair of functors $f_* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $f^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ such that

   1. we have

      \[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, f_*\mathcal{F}) \]

      bifunctorially, and

   2. the functor $f^{-1}$ commutes with finite limits, i.e., is left exact.

2. Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be sites. Given morphisms of topoi $f :\mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $g :\mathop{\mathit{Sh}}\nolimits (\mathcal{E}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ the composition $f\circ g$ is the morphism of topoi defined by the functors $(f \circ g)_* = f_* \circ g_*$ and $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$.