Remark 4.8.7.19 (05CK)

Remark 4.8.7.19. In the situation of Corollary 4.8.7.18, suppose that one of the following additional conditions is satisfied:

$(a)$: The functor $F$ is a monomorphism of simplicial sets.
$(b)$: The functor $G$ is an isofibration.

Condition $(a)$ guarantees that the horizontal maps in the diagram (4.86) are isofibrations (Corollary 4.4.5.3) and condition $(b)$ guarantees that the vertical maps are isofibrations (Corollary 4.4.5.6). In either case, the conclusion of Corollary 4.8.7.18 is equivalent to the requirement that the functor

\[ V: \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{D}}) } \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \]

is an equivalence of $\infty $-categories (Proposition 4.5.2.26). If conditions $(a)$ and $(b)$ are both satisfied, then $G$ is an isofibration of $\infty $-categories (Proposition 4.4.5.1). In this case, the conclusion of Corollary 4.8.7.18 is equivalent to the requirement that $G$ is a trivial Kan fibration (Proposition 4.5.5.20).