Corollary 9.4.6.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. The following conditions are equivalent:
- $(1)$
The pullback functor
\[ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{\mathcal{E}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]preserves Morita equivalences of simplicial sets. That is, if $F: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is a Morita equivalence in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$, then the induced map $F_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}'' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is also a Morita equivalence.
- $(2)$
If $F: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is a categorical equivalence in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$, then $F_{\operatorname{\mathcal{E}}}$ is a Morita equivalence.
- $(3)$
If $F: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is a categorical equivalence in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ which is surjective on vertices, then $F_{\operatorname{\mathcal{E}}}$ is a categorical equivalence.
- $(4)$
The inner fibration $U$ is flat.