Proof.
We proceed as in the proof of Proposition 7.5.4.10. Using Proposition 4.1.3.2, we can choose a functor $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a natural transformation $\beta : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ for which the morphism of simplicial sets $\beta _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {G}}(C)$ is inner anodyne for each object $C \in \operatorname{\mathcal{C}}^{\triangleleft }$. We will show that $(1)$ and $(2)$ are equivalent to the following:
- $(3)$
The functor $\overline{\mathscr {G}}$ is a categorical limit diagram.
The implications $(2) \Rightarrow (3) \Rightarrow (1)$ are immediate. To prove the reverse implications, suppose we are given another functor $\overline{\mathscr {F}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {F}}'$. We will show that $\overline{\mathscr {F}}'$ is a categorical limit diagram if and only if $\overline{\mathscr {G}}$ is a categorical limit diagram.
Applying Proposition 4.1.3.2 again, we can choose a functor $\overline{\mathscr {G}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}} \ar [r]^-{\alpha } \ar [d]^{\beta } & \overline{\mathscr {F}}' \ar [d]^{\beta '} \\ \overline{\mathscr {G}} \ar [r]^-{\alpha '} & \overline{\mathscr {G}}' } \]
with the property that, for every object $C \in \operatorname{\mathcal{C}}^{\triangleleft }$, the induced morphism of simplicial sets $\overline{\mathscr {F}}'(C) {\coprod }_{ \overline{\mathscr {F}}(C) } \overline{\mathscr {G}}(C) \rightarrow \overline{\mathscr {G}}'(C)$ is inner anodyne (and, in particular, a categorical equivalence). Applying Proposition 4.5.4.11, we see that the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}}(C) \ar [r]^-{\alpha _ C} \ar [d]^{\beta _ C} & \overline{\mathscr {F}}'(C) \ar [d]^{\beta '_{C}} \\ \overline{\mathscr {G}}(C) \ar [r]^-{\alpha '_{C}} & \overline{\mathscr {G}}'(C) } \]
is a categorical pushout square. Since $\alpha _ C$ and $\beta _ C$ are categorical equivalences, it follows that $\alpha '_ C$ and $\beta '_{C}$ are also categorical equivalences (Proposition 4.5.4.10). Applying Remark 7.5.5.6, we see that $\overline{ \mathscr {F} }'$ and $\overline{ \mathscr {G} }$ are categorical limit diagrams if and only if $\overline{ \mathscr {G} }'$ is a categorical limit diagram.
$\square$