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Remark 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 to the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$.