Example 7.5.2.14. Let $I$ be a set, which we regard as a category having only identity morphisms. Let $\mathscr {F}: I \rightarrow \operatorname{QCat}$ be a diagram, which we view as a collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ indexed by $I$. Then the comparison morphism
\[ {\prod }_{i \in I} \operatorname{\mathcal{C}}_ i = \varprojlim (\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \]
of Remark 7.5.2.12 is an isomorphism.