$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $f: A \hookrightarrow B$ be a right anodyne morphism of simplicial sets. Then the induced map

\[ \theta : B \coprod _{A} (A \diamond \Delta ^0) \hookrightarrow B \diamond \Delta ^0 \]

is a categorical equivalence of simplicial sets.

Proof. Proposition guarantees that the natural map $B \coprod _{A} A^{\triangleright } \hookrightarrow B^{\triangleright }$ is inner anodyne, and therefore a categorical equivalence (Corollary Using Proposition, we conclude that the diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A^{\triangleright } \ar [d]^{f^{\triangleright }} \\ B \ar [r] & B^{\triangleright } } \]

is categorical pushout square. It then follows from Theorem and Proposition that the equivalent diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A \diamond \Delta ^0 \ar [d]^{f \diamond \operatorname{id}_{\Delta ^{0}}} \\ B \ar [r] & B \diamond \Delta ^{0} } \]

is also categorical pushout square, so that $\theta $ is a categorical equivalence by virtue of Proposition $\square$