Remark 7.5.4.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $\overline{\mathscr {F}}$ is a homotopy limit diagram in the sense of Definition 7.5.4.1 (that is, it induces a homotopy equivalence $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}} )$, where ${\bf 0}$ denotes the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$).
- $(2)$
The functor $\overline{\mathscr {F}}$ is a homotopy limit diagram in the sense of Definition 7.5.4.8: that is, there exists a homotopy limit diagram of Kan complexes $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ and a levelwise weak homotopy equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$.
The implication $(1) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 7.5.4.4.