Proposition 7.6.4.1. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]
be a commutative diagram in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The composite map
\[ \Delta ^1 \times \Delta ^1 \xrightarrow { \operatorname{N}_{\bullet }(\sigma ) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \]is a pullback square in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (in the sense of Definition 7.6.3.1).
- $(2)$
For every object $Y \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X_{01} )_{\bullet } \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, X_0 )_{\bullet } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, X_1 )_{\bullet } \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)_{\bullet } } \]is a homotopy pullback square (in the sense of Definition 3.4.1.1).