The Stacks project

Lemma 7.25.5. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ commutes with fibre products and equalizers (and more generally finite connected limits). In particular, if $\mathcal{F} \subset \mathcal{F}'$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$, then $j_{U!}\mathcal{F} \subset j_{U!}\mathcal{F}'$.

Proof. Via Lemma 7.25.4 and the fact that an equivalence of categories commutes with all limits, this reduces to the fact that the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \rightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ commutes with fibre products and equalizers. Alternatively, one can prove this directly using the description of $j_{U!}$ in Lemma 7.25.2 using that sheafification is exact. (Also, in case $\mathcal{C}$ has fibre products and equalizers, the result follows from Lemma 7.21.6.) $\square$

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