Lemma 9.1.10.6. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be the colimit of a diagram of simplicial sets $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ indexed by a $\kappa $-directed partially ordered set $(A, \leq )$. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is obtained as the colimit of a compatible system of morphisms $F_{\alpha }: K \rightarrow \operatorname{\mathcal{C}}_{\alpha }$. If $K$ is $\kappa $-small, then the canonical map $\theta _{\bullet }: \varinjlim _{\alpha } ( \operatorname{\mathcal{C}}_{\alpha } )_{ F_{\alpha } / } \rightarrow \operatorname{\mathcal{C}}_{F/}$ is an isomorphism of simplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix an integer $n \geq 0$; we wish to show that $\theta _{\bullet }$ is bijective on simplices of dimension $n$. Unwinding the definitions, we see that $\theta _{n}$ is obtained from a commutative diagram of sets
\[ \xymatrix@R =50pt@C=50pt{ \varinjlim _{\alpha } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K \star \Delta ^ n, \operatorname{\mathcal{C}}_{\alpha } ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K \star \Delta ^ n, \operatorname{\mathcal{C}}) \ar [d] \\ \varinjlim _{\alpha } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{\mathcal{C}}_{\alpha } ) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{\mathcal{C}}) } \]
by passing to fibers in the vertical direction. It will therefore suffice to show that the horizontal maps are bijective. Since $(A, \leq )$ is $\kappa $-directed, this follows from the fact that $K$ and $K \star \Delta ^ n$ are $\kappa $-small simplicial sets (see Variant 9.2.2.10). $\square$