Remark 7.1.4.8 (05QC)

Remark 7.1.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let

\[ q: K \rightarrow \operatorname{\mathcal{C}}\quad \quad k \mapsto C_ k \]

be a diagram which admits a limit $C = \varprojlim _{k \in K} C_ k$. Suppose we are given a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ such that the diagram $(F \circ q): K \rightarrow \operatorname{\mathcal{D}}$ admits a limit $D = \varprojlim _{k \in K} F(C_ k)$. Choose natural transformations

\[ \alpha : \underline{C} \rightarrow q \quad \quad \beta : \underline{D} \rightarrow F \circ q \]

which exhibit $C$ as a limit of $q$ and $D$ as a limit $F \circ q$, respectively. Invoking the universal property of $\beta $, we see that there is a morphism $\gamma : F(C) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$ and a commutative diagram

\[ \xymatrix { F( \underline{C} ) \ar [rr]^{ \underline{\gamma } } \ar [dr]_{ F(\alpha ) } & & \underline{D} \ar [dl]^{ \beta } \\ & F \circ q & } \]

in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$; moreover, $\gamma $ is unique up to homotopy. Stated more informally, there is a natural comparison map

\[ F( \varprojlim _{k \in K} C_ k ) \rightarrow \varprojlim _{k \in K} F( C_ k ), \]

which is an isomorphism if and only if the functor $F$ preserves the limit of $q$.