$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram which admits a limit in $\operatorname{\mathcal{C}}$. Choose an object $X \in \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \underline{X} \rightarrow q$ which exhibits $X$ as a limit of $q$. Then $F$ preserves the limit of $q$ if and only if the natural transformation $F(\alpha )$ exhibits the object $F(X)$ as a limit of the diagram $F \circ q$.