Remark 7.1.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $Y \in \operatorname{\mathcal{C}}$ be an object equipped with a natural transformation $\alpha : \underline{Y} \rightarrow u$. If $F(\alpha ): \underline{F(Y)} \rightarrow (F \circ u)$ exhibits $F(Y)$ as a limit of the diagram $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$, then $\alpha $ exhibits $Y$ as a limit of $u$. The converse holds if $F$ is an equivalence of $\infty $-categories.
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