Lemma 7.10.13. Let $\mathcal{F} : \mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a diagram. Then $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} \mathcal{F}$ exists and is the sheafification of the colimit in the category of presheaves.

** Colimit in category of sheaves equals sheafification of colimit in category of presheaves. **

**Proof.**
Since the sheafification functor is a left adjoint it commutes with all colimits, see Categories, Lemma 4.24.5. Hence, since $\textit{PSh}(\mathcal{C})$ has colimits, we deduce that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ has colimits (which are the sheafifications of the colimits in presheaves).
$\square$

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## Comments (1)

Comment #7344 by Alejandro González Nevado on

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