$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 6.3.2.12. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets which is given as the colimit (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$) of a filtered diagram of morphisms $\{ G_{\alpha }: \operatorname{\mathcal{D}}_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\alpha } \} $. Assume that:
Each morphism $G_{\alpha }$ exhibits $\operatorname{\mathcal{E}}_{\alpha }$ as a localization of $\operatorname{\mathcal{D}}_{\alpha }$ with respect to some collection of edges $S_{\alpha }$.
Each of the transition maps $\operatorname{\mathcal{D}}_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\beta }$ of the diagram carries $S_{\alpha }$ into $S_{\beta }$.
Let us regard $S = \varinjlim S_{\alpha }$ as a collection of edges of the simplicial set $\operatorname{\mathcal{D}}$. Then $G$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{D}}$ with respect to $S$.
Proof.
For every simplicial set $\operatorname{\mathcal{C}}$ equipped with a set of edges $W$, let $F_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ be as in Proposition 6.3.2.11. For each index $\alpha $, let $T_{\alpha }$ denote the image of $S_{\alpha }$ in $\operatorname{\mathcal{E}}_{\alpha }$. We then have a commutative diagram of simplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{D}}_{\alpha } \ar [r]^{G_{\alpha }} \ar [d]^{ F_{\operatorname{\mathcal{D}}_{\alpha }}} & \operatorname{\mathcal{E}}_{\alpha } \ar [d]^{ F_{\operatorname{\mathcal{E}}_{\alpha }} } \\ \operatorname{\mathcal{D}}_{\alpha }[ S_{\alpha }^{-1} ] \ar [r] & \operatorname{\mathcal{E}}_{\alpha }[ T_{\alpha }^{-1} ], } \]
depending functorially on $\alpha $. Our assumption that $G_{\alpha }$ exhibits $\operatorname{\mathcal{E}}_{\alpha }$ as a localization of $\operatorname{\mathcal{D}}_{\alpha }$ with respect to $S_{\alpha }$ guarantees that the lower horizontal and right vertical maps are categorical equivalences of simplicial sets. Setting $U = G(T)$ and passing to the colimit over $\alpha $, we obtain a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{D}}\ar [r]^{G} \ar [d]^{F_{\operatorname{\mathcal{D}}}} & \operatorname{\mathcal{E}}\ar [d]^{ F_{\operatorname{\mathcal{E}}} } \\ \operatorname{\mathcal{D}}[ S^{-1} ] \ar [r] & \operatorname{\mathcal{E}}[ T^{-1} ] } \]
where the lower horizontal and right vertical maps are categorical equivalences (Corollary 4.5.7.2). It follows that $G$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{D}}$ with respect to $S$.
$\square$