Lemma 7.21.5. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that
$u$ is cocontinuous, and
$u$ is continuous.
Sheafification is redundant in topoi morphisms associated to simultaneously continuous and cocontinuous site functors.
Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the associated morphism of topoi. Then
sheafification in the formula $g^{-1} = (u^ p\ )^\# $ is unnecessary, in other words $g^{-1}(\mathcal{G})(U) = \mathcal{G}(u(U))$,
$g^{-1}$ has a left adjoint $g_{!} = (u_ p\ )^\# $, and
$g^{-1}$ commutes with arbitrary limits and colimits.
Proof. By Lemma 7.13.2 for any sheaf $\mathcal{G}$ on $\mathcal{D}$ the presheaf $u^ p\mathcal{G}$ is a sheaf on $\mathcal{C}$. And then we see the adjointness by the following string of equalities
\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G}) & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^ p\mathcal{G}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(u_ p\mathcal{F}, \mathcal{G}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(g_{!}\mathcal{F}, \mathcal{G}) \end{eqnarray*}
The statement on limits and colimits follows from the discussion in Categories, Section 4.24. $\square$
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Comment #7347 by Alejandro González Nevado on