The Stacks project

Lemma 18.16.6. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$.

  1. If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\mathop{\mathit{Sh}}\nolimits }$ on underlying sheaves of sets and $g_!$ is exact.

  2. If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is the morphism of topoi associated to $w$.

Proof. This Lemma is the analogue of Sites, Lemma 7.23.1. From Sites, Lemma 7.19.3 we see that the categories $\mathcal{I}_ V^ u$ have an initial object. Thus the underlying set of a colimit of a system of abelian groups over $(\mathcal{I}_ V^ u)^{opp}$ is the colimit of the underlying sets. Whence the agreement of $g_!^{\mathop{\mathit{Sh}}\nolimits }$ and $g_!$ by our construction of $g_!$ in Definition 18.16.1. The exactness and (2) follow immediately from the corresponding statements of Sites, Lemma 7.23.1. $\square$

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