Corollary 7.6.4.14. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and suppose we are given a commutative diagram $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ \xymatrix@C =100pt@R=100pt{ X_{01} \ar [r]^-{g_0} \ar [d]_-{g_1} \ar [dr]^-{h} & X_0 \ar [d]^-{f_0} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>_-{\alpha _0} \\ X_1 \ar [r]_-{ f_1 } \ar@ {=>}[]+<20pt,20pt>;+<45pt,45pt>^-{\alpha _1} & X } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Remark 7.6.4.6). Then $\sigma $ is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Y \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X_{01})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X_0)_{\bullet } \times ^{\mathrm{h}}_{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } } ( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X_1)_{\bullet } \times ^{\mathrm{h}}_{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } }\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } ) \]
is a homotopy equivalence of Kan complexes.