I have a question about a proof from Jacob Lurie's Higher Topos Theory (p 45):

**Proposition 1.2.12.5.** Let $\mathcal{C}$ be a simplicial set. Every strongly final object (see below what means) of
$\mathcal{C}$ is a final object of $\mathcal{C}$. The converse holds if $\mathcal{C}$ is an $\infty$-category.

Proof:Let $[0]$ denote the category with a single object and a single morphism. Suppose that $Y$ is a strongly final vertex of $\mathcal{C}$. Then there exists a retraction of $\mathcal{C}^{ \triangleright }$ onto $\mathcal{C}$ carrying the cone point to $Y$ . Consequently, we obtain a retraction of (H-enriched) homotopy categories from $h\mathcal{C} * [0]$ to $h\mathcal{C}$ carrying the unique object of $[0]$ to $Y$ . This implies that $Y$ is final in $h\mathcal{C}$, so that $Y$ is a final object of $\mathcal{C}$.

(The converse part in the proof I understand)

On used notations:

-$\mathcal{C}^{ \triangleright }$ is the *right cone* defined to be the join $\mathcal{C} * \Delta^0$.

-*Definition 1.2.12.3.:* A vertex $X$ of simplicial set $\mathcal{C}$ is called *strongly final* if the projection $ \mathcal{C}_{/ X} \to \mathcal{C}$ is a trivial fibration of simplicial sets. In other words if any map $f_0 : \partial \Delta^n \to \mathcal{C}$ such that $f_0(n) = X$ can be extended to a map $f : \Delta^n \to \mathcal{C}$.

*Question:* Why does the assumption that $Y$ is a strongly final vertex of $\mathcal{C}$ imply that there exists a
retraction of $\mathcal{C}^{ \triangleright }$ onto $\mathcal{C}$ carrying the cone point to $Y$? Can its construction be given explicitly or (if not) which abstract result is involved to deduce the existence of this retraction?