Corollary 8.5.8.9. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a filtered category $\operatorname{\mathcal{I}}$. Suppose that each $\operatorname{\mathcal{C}}_{i}$ is an $\infty $-category. Then the tautological map
\[ \theta : \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ) \]
is an equivalence of $\infty $-categories.
Proof.
Choose any integer $n \geq 3$, and let $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$, $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, and $V: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ be as in Theorem 8.5.8.4. We then have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \ar [r]^-{\circ V} \ar [d]^{\theta } & \varinjlim \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}_ i ) \ar [d]^{\theta '} \ar [r]^-{ \circ U} & \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}) , \operatorname{\mathcal{C}}_{i} ) \ar [d]^{\theta } \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \varinjlim \operatorname{\mathcal{C}}_{i}) \ar [r]^-{ \circ V} & \operatorname{Fun}( \operatorname{\mathcal{E}}, \varinjlim \operatorname{\mathcal{C}}_{i} ) \ar [r]^-{ \circ U} & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \varinjlim \operatorname{\mathcal{C}}_{i} ), } \]
where the horizontal compositions are identity morphisms. Consequently, to show that $\theta $ is an equivalence of $\infty $-categories, it will suffice to show that $\theta '$ is an equivalence of $\infty $-categories (Proposition 8.5.1.7). The functor $\theta '$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \varinjlim \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}_ i ) \ar [r]^-{ \circ \iota } \ar [d]^{\theta '} & \varinjlim \operatorname{Fun}( \operatorname{N}_{\leq n}(\operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \ar [d]^{\theta ''} \\ \operatorname{Fun}( \operatorname{\mathcal{E}}, \varinjlim \operatorname{\mathcal{C}}_ i ) \ar [r]^-{ \circ \iota } & \operatorname{Fun}( \operatorname{N}_{\leq n}( \operatorname{Idem}), \varinjlim (\operatorname{\mathcal{C}}_ i) ). } \]
Since $\iota $ is inner anodyne, the horizontal maps are trivial Kan fibrations (Proposition 1.5.7.6). We conclude by observing that $\theta ''$ is an isomorphism of simplicial sets, since the simplicial set $\operatorname{N}_{\leq n}( \operatorname{Idem})$ is finite (Corollary 3.6.1.10).
$\square$