Exercise 7.1.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $K$ be a simplicial set. Show that $F$ preserves $K$-indexed limits if and only if it satisfies the following condition:
For every diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ and every natural transformation $\alpha : \underline{Y} \rightarrow u$ which exhibits an object $Y \in \operatorname{\mathcal{C}}$ as a limit of $u$ (in the sense of Definition 7.1.1.1), the image $F(\alpha ): \underline{F(Y)} \rightarrow (F \circ u)$ exhibits the object $F(Y) \in \operatorname{\mathcal{D}}$ as a limit of the diagram $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$.