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Remark Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism, and set $f = \overline{f}|_{K}$, so that $U$ induces a functor

\[ U': \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f} \times _{ \operatorname{\mathcal{D}}_{/(U \circ f)}} \operatorname{\mathcal{D}}_{ / (U \circ \overline{f} ) }. \]

By virtue of Proposition, the following conditions are equivalent:


The morphism $\overline{f}$ is a $U$-limit diagram.


The functor $U'$ is an equivalence of $\infty $-categories.

If $U$ is an inner fibration of $\infty $-categories, then the functor $U'$ is automatically a right fibration (Proposition In this case, we can replace $(1)$ and $(2)$ by either of the following conditions: to the following:


The functor $U'$ is a trivial Kan fibration.


Each fiber of $U'$ is a contractible Kan complex.

The equivalence of $(2) \Leftrightarrow (3)$ follows from Proposition, and the equivalence $(3) \Leftrightarrow (4)$ from Proposition