Example 5.2.4.6. In the special case $Y = \Delta ^0$, Proposition 5.2.4.5 asserts that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times X \ar [r] \ar [d] & \Delta ^{0} \ar [d] \\ \Delta ^1 \times X \ar [r] & X^{\triangleright } } \]
is a categorical pushout square: that is, that the comparison map $X \diamond \Delta ^{0} \rightarrow X \star \Delta ^{0}$ of Notation 4.5.8.3 is a categorical equivalence. This is the content of Proposition 4.5.8.12 (which is a special case of Theorem 4.5.8.8).