The Stacks project

Lemma 7.21.8. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that

1. $u$ is cocontinuous,

2. $u$ is continuous,

3. $u$ is fully faithful,

4. fibre products exist in $\mathcal{C}$ and $u$ commutes with them, and

5. there exist final objects $e_\mathcal {C} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $e_\mathcal {D} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ such that $u(e_\mathcal {C}) = e_\mathcal {D}$.

Let $g_!, g^{-1}, g_*$ be as above. Then, $u$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ with $f_* = g^{-1}$, $f^{-1} = g_!$. The composition

is isomorphic to the identity morphism of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Moreover, the functor $f^{-1}$ is fully faithful.