Remark 7.1.5.8. 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
By virtue of Proposition 7.1.4.16, the following conditions are equivalent:
- $(1)$
The morphism $\overline{f}$ is a $U$-limit diagram.
- $(2)$
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 4.3.6.8). In this case, we can replace $(1)$ and $(2)$ by either of the following conditions:
- $(3)$
The functor $U'$ is a trivial Kan fibration.
- $(4)$
Each fiber of $U'$ is a contractible Kan complex.
The equivalence of $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20, and the equivalence $(3) \Leftrightarrow (4)$ from Proposition 4.4.2.14.