Remark 7.5.5.12. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $\overline{\mathscr {F}}$ is a categorical limit diagram in the sense of Definition 7.5.5.1: that is, it induces an equivalence of $\infty $-categories $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}} )$.
- $(2)$
The functor $\overline{\mathscr {F}}$ is a categorical limit diagram in the sense of Definition 7.5.5.11: that is, there exists a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$, where $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ induces an equivalence of $\infty $-categories $\overline{\mathscr {G}}( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {G}}|_{\operatorname{\mathcal{C}}} )$.
The implication $(1) \Rightarrow (2)$ is immediate, and the reverse implication follows from Remark 7.5.5.6.