Construction 9.2.7.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma :$
be a lifting problem in $\operatorname{\mathcal{C}}$. Then $\sigma $ determines a pair of objects $\widetilde{B}$, $\widetilde{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{A/ \, / Y }$ (see Remark 9.2.5.7). Let $K$ denote the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X} )$. We then have a tautological map $K \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}_{A/ \, /Y}$, which we can identify with a diagram $\{ A\} \star (K \times \Delta ^1) \star \{ Y\} \rightarrow \operatorname{\mathcal{C}}$. Composing with the quotient map
we obtain a morphism $K \rightarrow \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}})$, which factors through the simplicial subset $\operatorname{Sol}(\sigma ) \subseteq \operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})$ of Construction 9.2.7.1. We therefore obtain a comparison map $\theta : \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X} ) \rightarrow \operatorname{Sol}( \sigma )$.