Lurie introduced in subchapter 1.2.12 of his [Higher Topos Theory][1]
the notion of *final* and *strongly final* objects:

**Definition 1.2.12.1.** let $\mathcal{C}$ be a topological category (e.g. simplicial cats,
simplicial set). 
An object $X \in \mathcal{C}$ is *final* if for each $Y \in \mathcal{C}$,
the mapping space $\text{Map}_{h\mathcal{C}}(Y,X)$
regarded itself as object in associated homotopy category $h\mathcal{C}$
 is *weakly contractible*, that is a final object (in ordinary category sense) in  $h\mathcal{C}$.

**Definition 1.2.12.3.** Let $\mathcal{C}$ be now a simplicial set. 
An object ("vertex") $X$
 of $\mathcal{C}$ is strongly
final if the projection $p: \mathcal{C}/X \to \mathcal{C} $ is a 
trivial fibration of simplicial sets.

The used slice $\mathcal{C}/X$ for a vertex $X: \Delta^0 \to C$ is a simplicial set
whose $n$-simplices are $f \in \text{Hom}_X(\Delta^n \ast \Delta^0,C)$
where the subscript $X$ says that these are subjected to condition
$f \vert _{\Delta^0} =X$ (Proposition 1.2.9.2)


Two questions about some properties of these definitions:

1) Why is object $X \in \mathcal{C}$ final if there is a *retraction*
 of ($h\mathcal{C}$-enriched)
 homotopy categories from $h\mathcal{C} \ast [0]$ to 
$h\mathcal{C}$ carrying the unique object of $[0]$ to $X$. In other words
why the existence of such retraction implies that 
for each $Y \in \mathcal{C}$,
the mapping space $\text{Map}_{h\mathcal{C}}(Y,X)$ is 
*weakly contractible* in above sense. (this statement
in used in the proof of Corollary 1.2.12.5)

2) After *Definition 1.2.12.3.* of strongly
final vertex $X$ is remarked that it's equivalent to that vertex
$X \in \mathcal{C}$ is *strongly final* if and only if any map
$f_0: \partial \Delta^n \to \mathcal{C}$ such that $f_0(n) =X$ 
can be lifted to a map $f: \Delta^n \to \mathcal{C}$.

Why that' true? Applying Definition 1.2.12.3 above $X$ is strongly final if the natural
 projection $p: \mathcal{C}/X \to \mathcal{C}$ is a trivial fibration of simplicial sets.
In other words $p: \mathcal{C}/X \to \mathcal{C}$ has the
lifting property with respect to every inclusion
$\partial \Delta^n \subset \Delta^n $. This means that
the condition should be that read as that 
a map $f_0: \partial \Delta^n \to \mathcal{C}/X$ such that 
$p \circ f_0: \partial \Delta^n \to \mathcal{C}$ extends to 
$\overline{f}: \Delta^n \to \mathcal{C}$, extends to
a $f: \Delta^n \to \mathcal{C}/X$ with $p \circ f=\overline{f}$.


Equivalently using adjuncion from defining property of slice
$\mathcal{C}/X$ we have

$$  \text{Hom}_{sSet}(S,\mathcal{C}/X) = 
\text{Hom}_X(S \ast \Delta^0,\mathcal{C}) $$ 

for any simplicial set $S$, the lifting property can be reformulated 
in terms of lifting a 
$f_0: \partial \Delta^n \ast \Delta^0 \to \mathcal{C}$
with $f_0 \vert _{\Delta^0} =X$ to a 
$f: \Delta^n \ast \Delta^0 \to \mathcal{C}$. But this lifting 
property is seemingly also not the same as stated as remark 
after Definition 1.2.12.3 in the book. Or is it in some nested sense?


  [1]: https://arxiv.org/abs/math/0608040