I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help.

First, in Lemma 2.2.3.6, while proving $(a)\implies (c)$, Lurie constructed a class $\scr U$ of simplicial sets which contains all simplicial sets $A$ satisfying $(c)$. The author claimed that using Lemma 2.2.3.4, it can be shown that the condition $(e)$ of Lemma 2.2.3.5 is satisfied. But I have no idea how Lemma 2.2.3.4 can be used here, nor can I figure out any other proof. How should I use Lemma 2.2.3.4 to prove this?

Second, in Corollary 2.2.3.12, Lurie claimed that for the simplicial equivalence $f:X\to Y$ in ${\sf SSet}_{/S}$ constructed there exists a map $g:Y\to X$ in ${\sf SSet}_{/S}$ and a homotopy $h:X\times\Delta[1]\to X$ from $\mathbb1_X$ to $g\circ f$. Such claim is true in a topological category, but in my thoughts two maps $f_1,f_2:A\to B$ in a simplicial category coincides in the homotopy category does not mean that there exists a homotopy $h:A\times\Delta[1]\to B$ connecting $f_1,f_2$. Is my thought incorrect, or the addition assumptions to $X,Y$ ($X\to S$ is a right fibration and $Y$ is contravariantly fibrant) may ensure that such $h$ does exist?