Lurie introduced in subchapter 1.2.12 of his Higher Topos Theory 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?

Lurie introduced in subchapter 1.2.12 of his Higher Topos Theory 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 if $p: \mathcal{C}/X \to \mathcal{C}$ has the lifting property with respect to every inclusion $\partial \Delta^n \subset \Delta^n $. This means that the condition should be 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 the 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 the remark after Definition 1.2.12.3 in the book I exposed in point 2. Or is it in some implicit sense?