Remark 4.6.4.4 (Homotopy Invariance). Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [d] & \operatorname{\mathcal{D}}\ar [l] \ar [d] \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{E}}' & \operatorname{\mathcal{D}}', \ar [l] } \]
where the vertical maps are equivalences of $\infty $-categories. Then the induced map
\[ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}' \]
is also an equivalence of $\infty $-categories. This follows by applying Corollary 4.5.2.30 to the diagram
\[ \xymatrix { \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}) \ar [d] & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [l] \ar [d] \\ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}}') \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}') & \operatorname{\mathcal{C}}' \times \operatorname{\mathcal{D}}'. \ar [l] } \]