Remark 4.5.4.3. Suppose we are given a categorical pushout square of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01}. } \]
Then, for every simplicial set $K$, the induced diagram
\[ \xymatrix@R =50pt@C=50pt{ A \times K \ar [r] \ar [d] & A_0 \times K\ar [d] \\ A_1 \times K \ar [r] & A_{01} \times K } \]
is also a categorical pushout square. That is, for every $\infty $-category $\operatorname{\mathcal{C}}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A \times K,\operatorname{\mathcal{C}})^{\simeq } & \operatorname{Fun}(A_0 \times K,\operatorname{\mathcal{C}})^{\simeq } \ar [l] \\ \operatorname{Fun}(A_1 \times K, \operatorname{\mathcal{C}})^{\simeq } \ar [u] & \operatorname{Fun}(A_{01} \times K, \operatorname{\mathcal{C}})^{\simeq } \ar [u] \ar [l] } \]
is a homotopy pullback square. This follows by applying the requirement Definition 4.5.4.1 to the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$.