Lemma 9.2.8.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Then the comparison map
of Notation 9.2.8.5 is a homotopy equivalence.
Proof. By construction, the diagram $\widetilde{f}: \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from the simplicial subset $\Delta ^1 \subset \Delta ^2$. Applying Corollary 7.3.6.9, we deduce that the restriction functor $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ determines a trivial Kan fibration $R: \operatorname{Hom}_{\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} )$. Similarly, the $1$-simplex $\operatorname{id}_{X}$ can be viewed as a diagram $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ which is right Kan extended from the vertex $\{ 1\} \subset \Delta ^2$, so the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ induces a trivial Kan fibration $Q: \operatorname{Hom}_{\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B, X )$. We are therefore reduced to showing that the composite map
is a homotopy equivalence, which follows from Lemma 9.2.8.4. $\square$