Remark 9.2.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Fix a morphism $\overline{u}_0: A \rightarrow Y$ of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}_{ A/ \, /Y}$ denote the $\infty $-category of Remark 4.6.6.2. The datum of an extension of $\overline{u}_0$ to a lifting problem $\sigma :$
9.14
\begin{equation} \begin{gathered}\label{equation:lifting-problem-via-double-slice} \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & X \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & Y } \end{gathered} \end{equation}
can be identified with a pair of objects $\widetilde{B}$, $\widetilde{X} \in \operatorname{\mathcal{C}}_{A/ \, /Y}$. In this case, a solution to the lifting problem (9.14) is a morphism from $\widetilde{B}$ to $\widetilde{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{A/ \, /Y}$.