# Kerodon

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Corollary 4.5.8.14. 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 4.3.6.4 guarantees that the natural map $B \coprod _{A} A^{\triangleright } \hookrightarrow B^{\triangleright }$ is inner anodyne, and therefore a categorical equivalence (Corollary 4.5.3.14). Using Proposition 4.5.4.11, 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 4.5.8.8 and Proposition 4.5.4.9 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 4.5.4.11. $\square$