Corollary 7.5.5.14 (03C1)

Corollary 7.5.5.14. Let $\operatorname{\mathcal{C}}$ be a category, let $\overline{\mathscr {F}}, \overline{\mathscr {F}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be functors, and let $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {F}}'$ be a natural transformation. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {F}}'(C)$ is a categorical equivalence of simplicial sets. Then any two of the following conditions imply the third:

$(1)$: The functor $\overline{\mathscr {F}}$ is a categorical limit diagram.
$(2)$: The functor $\overline{\mathscr {F}}'$ is a categorical limit diagram.
$(3)$: The natural transformation $\alpha $ induces a categorical equivalence of simplicial sets $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \overline{\mathscr {F}}'( {\bf 0} )$, where ${\bf 0}$ denotes the initial object of $\operatorname{\mathcal{C}}^{\triangleleft }$.

Proof. Using Proposition 4.1.3.2, we can choose functors $\overline{\mathscr {G}}, \overline{\mathscr {G}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}} \ar [r]^-{\alpha } \ar [d] & \overline{\mathscr {F}}' \ar [d] \\ \overline{\mathscr {G}} \ar [r]^-{\beta } & \overline{\mathscr {G}}', } \]

where the vertical maps are levelwise categorical equivalences. By virtue of Proposition 7.5.5.13, we can replace $\alpha $ by the natural transformation $\beta : \overline{\mathscr {G}} \rightarrow \overline{\mathscr {G}}'$. In this case, the desired result follows from Remark 7.5.5.6. $\square$