Corollary 7.6.4.16. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and suppose we are given morphisms
\[ \xymatrix@R =50pt@C=50pt{ Z \ar [r]^-{g} & Y \ar@ <.4ex>[r]^-{f_0} \ar@ <-.4ex>[r]_-{f_1} & X } \]
in $\operatorname{\mathcal{C}}$ satisfying $f_0 \circ g = f_1 \circ g$. The following conditions are equivalent:
- $(1)$
The induced diagram $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an equalizer diagram in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, in the sense of Definition 7.6.4.10.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Z)_{\bullet } \ar [r] \ar [d]^-{g} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y )_{\bullet } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } \ar [r]^-{(f_0, f_1) } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } } \]is a homotopy pullback square.