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

Proposition (Rewriting Limits as Products). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\{ f_ i: K_ i \rightarrow \operatorname{\mathcal{C}}\} _{i \in I}$ be a collection of diagrams, each of which admits a limit $X_ i = \varprojlim (f_ i)$. Set $K = \coprod _{i \in I} K_ i$, so that the collection $\{ f_ i \} _{i \in I}$ determines a diagram $f: K \rightarrow \operatorname{\mathcal{C}}$. Then an object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $f$ if it is a product of the collection of objects $\{ X_ i \} _{i \in I}$.

Proof. This is a special case of (the dual of) Proposition $\square$