Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories

Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed.

Here is the correct statement of (*):

For any cofibration $f:Ato B$ and any trivial fibration $g:Xto Y$ in $C$, the induced morphism:

$$operatorname{Map}(B,X)to operatorname{Map}(B,Y)times_{operatorname{Map}(A,Y)} operatorname{Map}(A,X)$$

is a trivial Kan fibration. (Where $operatorname{Map}$ is the (sSet)-enriched $operatorname{Hom}$).

Let $C$ be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment.

Suppose that all objects of $C$ are cofibrant, that $C$ is tensored and cotensored over $SSet$, and that the class of weak equivalences of $C$ is closed under filtered colimits.

(*)Suppose further that for any cofibration $f:Ato B$ and any fibration $g:Xto Y$ in $C$, the induced morphism:

$$operatorname{Map}(B,X)to operatorname{Map}(B,Y)times_{operatorname{Map}(A,Y)} operatorname{Map}(A,X)$$

is a Kan fibration. (Where $operatorname{Map}$ is the (sSet)-enriched $operatorname{Hom}$).

Lastly, assume that $Aotimes Delta^ntilde{to}Aotimes Delta^0=A$ is a weak equivalence for any object $A$ in $C$ and any $nin mathbf{N}$. (Here, the tensor $Aotimes K$, where $A$ is in $C$ and $K$ is in $sSet$, is the object of $C$ representing the functor $Map(A,-)^K$).

Let $Lsubseteq K$ be an inclusion of simplicial sets. Suppose $sigma:Delta^nhookrightarrow K$ is a nondegenerate simplex of $K$ with all of its faces living in $L$. That is, we can factor the map $partialsigma:=sigma|_{partialDelta^n}$ through the inclusion $Lsubseteq K$ (in fact, we will assume that the target of this map actually is $L$).

Then for any object $D$ in $C$, the pushout $$Dotimes Delta^ncoprod_{DotimespartialDelta^n} Dotimes Lcong Dotimes (Delta^ncoprod_{partialDelta^n}L)$$ is a homotopy pushout. Now, the question here is, why is this the case?

The proof I'm reading says that it follows from the line marked (*) above (and the fact that $C$ is left-proper (which follows from the fact that all objects of $C$ are cofibrant)), but it's not clear to me how to apply that hypothesis.

That is, how does the line marked (*) imply anything relevant?

If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix).

Edit: It's probable that a few of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important.