Theorem 4.5.8.8. Let $X$ and $Y$ be simplicial sets. Then the comparison map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ of Notation 4.5.8.3 is a categorical equivalence of simplicial sets.
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Proof of Theorem 4.5.8.8. Let $X$ and $Y$ be arbitrary simplicial sets; we wish to show that the comparison map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ is a categorical equivalence. By virtue of Lemma 4.5.8.10, we may assume without loss of generality that $X = \Delta ^1$. Note that the map the $c_{X,Y}$ fits into a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \times Y \ar [r] \ar [d] & (X \diamond \Delta ^0) \times Y \ar [d] \ar [r]^-{c_{X,\Delta ^0} \times \operatorname{id}_ Y} & X^{\triangleright } \times Y \ar [d] \\ X \ar [r] & X \diamond Y \ar [r]^-{ c_{X,Y} } & X \star Y. } \]
Note that the morphism $c_{X,\Delta ^{0}} \times \operatorname{id}_{Y}$ is a categorical equivalence by virtue of Proposition 4.5.8.12 and Remark 4.5.3.7. Consequently, to show that $c_{X,Y}$ is a categorical equivalence, it will suffice to show that the square on the right is a categorical pushout (Proposition 4.5.4.10). Note that left part of the diagram is a pushout square in which the horizontal maps are monomorphisms, hence also a categorical pushout square (Proposition 4.5.4.11). We are therefore reduced to showing that the outer rectangle is a categorical pushout square (Proposition 4.5.4.8).
Specializing now to the case $X = \Delta ^1$, we wish to show that the lower part of the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times Y \ar [r] \ar [d] & \{ 1\} ^{\triangleright } \times Y \ar [d] \\ \Delta ^{1} \times Y \ar [r] \ar [d] & (\Delta ^1)^{\triangleright } \times Y \ar [d] \\ \Delta ^1 \ar [r] & \Delta ^1 \star Y } \]
is a categorical pushout square. We first claim that the upper square is a categorical pushout: by virtue of Proposition 4.5.4.11, this is equivalent to the assertion that the induced map
\[ \theta : (\Delta ^1 \times Y) \coprod _{ \{ 1\} \times Y} (\{ 1\} ^{\triangleright } \times Y) \rightarrow (\Delta ^1)^{\triangleright } \times Y \]
is a categorical equivalence. This follows from Remark 4.5.3.7, since $\theta $ factors as a product of the identity map $\operatorname{id}_{Y}$ with the inner horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. To complete the proof, it will suffice to show that the outer rectangle is a categorical pushout square. Using the criterion of Proposition 4.5.4.11, we are reduced to showing that the map
\[ \rho : \Delta ^1 \coprod _{ \{ 1\} \times Y} ( \{ 1\} ^{\triangleright } \times Y) \rightarrow \Delta ^1 \star Y \]
is a categorical equivalence. Unwinding the definitions, we can identify $\rho $ with the composition
\[ \Delta ^{1} \coprod _{ \{ 1\} } ( \{ 1\} \diamond Y ) \xrightarrow {\rho '} \Delta ^1 \coprod _{ \{ 1\} } ( \{ 1\} \star Y) \xrightarrow {\rho ''} \Delta ^1 \star Y. \]
Here the map $\rho '$ is a categorical equivalence by virtue of Proposition 4.5.8.12 (together with Remark 4.5.4.13), and the map $\rho ''$ is inner anodyne by virtue of Proposition 4.3.6.4. $\square$