Proposition 3.3.5.7. Let $Y$ be a Kan complex. Then the comparison map $\rho _{Y}: Y \rightarrow \operatorname{Ex}(Y)$ of Construction 3.3.4.3 is a homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix a simplicial set $X$. We wish to show that postcomposition with $\rho _{Y}$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Maps of simplicial sets $X \rightarrow Y$} \} / \text{homotopy} \ar [d] \\ \{ \text{Maps of simplicial sets $X \rightarrow \operatorname{Ex}(Y)$} \} / \text{homotopy}. } \]
By virtue of Proposition 3.3.5.6, this is equivalent to the assertion that precomposition with the last vertex map $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Maps of simplicial sets $X \rightarrow Y$} \} / \text{homotopy} \ar [d] \\ \{ \text{Maps of simplicial sets $\operatorname{Sd}(X) \rightarrow Y$} \} / \text{homotopy}, } \]
which follows from the fact that $\lambda _{X}$ is a weak homotopy equivalence (Proposition 3.3.4.8). $\square$