Lemma 4.27.9. Let \mathcal{C} be a category and let S be a left multiplicative system of morphisms of \mathcal{C}. The localization functor Q : \mathcal{C} \to S^{-1}\mathcal{C} commutes with finite colimits.
Proof. Let \mathcal{I} be a finite category and let \mathcal{I} \to \mathcal{C}, i \mapsto X_ i be a functor whose colimit exists. Then using (4.27.7.1), the fact that Y/S is filtered, and Lemma 4.19.2 we have
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(Q(\mathop{\mathrm{colim}}\nolimits X_ i), Q(Y)) & = \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathop{\mathrm{colim}}\nolimits X_ i, Y') \\ & = \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X_ i, Y') \\ & = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _{(s : Y \to Y') \in Y/S} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X_ i, Y') \\ & = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(Q(X_ i), Q(Y)) \end{align*}
and this isomorphism commutes with the projections from both sides to the set \mathop{\mathrm{Mor}}\nolimits _{S^{-1}\mathcal{C}}(Q(X_ j), Q(Y)) for each j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}). Thus, Q(\mathop{\mathrm{colim}}\nolimits X_ i) satisfies the universal property for the colimit of the functor i \mapsto Q(X_ i); hence, it is this colimit, as desired. \square
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