Lemma 1.2.3.6. Let $\operatorname{\mathcal{J}}$ be a small category equipped with a functor $S: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set_{\Delta }}$. Suppose that, for each $J \in \operatorname{\mathcal{J}}$, the simplicial set $S(J)$ admits a geometric realization $| S(J) |$. Then the colimit $T = \varinjlim _{J \in \operatorname{\mathcal{J}}} S(J)$ also admits a geometric realization, given by the colimit $Y = \varinjlim _{J \in \operatorname{\mathcal{J}}} | S(J) |$ in the category of topological spaces.
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Proof. For each $J \in \operatorname{\mathcal{J}}$, choose a topological space $| S(J) |$ and a map $u_{J}: S(J) \rightarrow \operatorname{Sing}_{\bullet }( | S(J) | )$ which exhibits $| S(J) |$ as a geometric realization of $S(J)$. We can then amalgamate the composite maps
\[ S(J) \xrightarrow {u_ J} \operatorname{Sing}_{\bullet }( | S(J) | ) \rightarrow \operatorname{Sing}_{\bullet }( Y ) \]
to a single map of simplicial sets $u: T \rightarrow \operatorname{Sing}_{\bullet }( Y)$. We claim that $u$ exhibits $Y$ as a geometric realization of the simplicial set $T$. Let $X$ be any topological space; we wish to show that the composite map
\[ \operatorname{Hom}_{ \operatorname{Top}}( Y, X ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet } (Y), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( T, \operatorname{Sing}_{\bullet }(X) ) \]
is a bijection. This is clear, since this composite map can be written as an inverse limit of the bijections $\operatorname{Hom}_{ \operatorname{Top}}( | S(J) |, X ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S(J), \operatorname{Sing}_{\bullet }(X) )$ determined by the maps $u_ J$. $\square$