Remark 4.5.8.7. Let $f: X \rightarrow X'$ and $g: Y \rightarrow Y'$ be categorical equivalences of simplicial sets. Then the induced map $(f \diamond g): X \diamond Y \rightarrow X' \diamond Y'$ is also a categorical equivalence. This follows by applying Proposition 4.5.4.9 to the diagram
\[ \xymatrix@C =20pt@R=50pt{ X \times \operatorname{\partial \Delta }^1 \times Y \ar [rr] \ar [dd] \ar [dr] & & X \times \Delta ^1 \times Y \ar [dd] \ar [dr] & \\ & X' \times \operatorname{\partial \Delta }^1 \times Y' \ar [rr] \ar [dd] & & X' \times \Delta ^1 \times Y' \ar [dd] \\ ( X \times \{ 0\} ) \coprod ( \{ 1\} \times Y ) \ar [rr] \ar [dr] & & X \diamond Y \ar [dr] & \\ & (X' \times \{ 0\} ) \coprod ( \{ 1\} \times Y' ) \ar [rr] & & X' \diamond Y'. } \]