Remark 4.5.2.7. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Applying Corollary 4.4.3.19 to the pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \ar [r] \ar [d] & \operatorname{Isom}(\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}, } \]
we deduce that the diagram of cores
\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1)^{\simeq } \ar [r] \ar [d] & \operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq } \ar [d] \\ \operatorname{\mathcal{C}}_0^{\simeq } \times \operatorname{\mathcal{C}}_{1}^{\simeq } \ar [r] & \operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq } } \]
is also a pullback square: that is, we have a canonical isomorphism of Kan complexes
\[ ( \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 )^{\simeq } \simeq \operatorname{\mathcal{C}}_0^{\simeq } \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}^{\simeq }} \operatorname{\mathcal{C}}_{1}^{\simeq }. \]