Proof of stability property of cartesian fibrations in Luries HTT I'm preparing a seminar lecture on $\infty$-categories. My main source is
Luries Higher Topos Theory, see:
http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf
I'm now trying to figure out the proof of Proposition 3.1.2.1, where he wants to use Proposition 3.1.2.3 to show, that
$p^K:(X^\natural)^{K^\flat}\rightarrow (S^K)^\sharp$
has the right lifting property (rlp) to all marked anodyne maps. I don't know how he wants to achieve this by using the property of smash-products of Proposition 3.1.2.3. 
I'm guessing there is some commutative cube, which "produces" the asserted lift for the rlp by using some marked anodyne map as in Proposition 3.1.2.3, and the fact that
$p:X^\natural\rightarrow S^\sharp$
has the rlp for all marked anodyne maps, but I dont know how to construct this cube, or if this is even the right approach to do it.

$\require{AMScd}$
If $f : X \to X'$ is marked anodyne and $Y'$ is a marked simplicial set, Proposition 3.1.2.3 claims that $$X \times Y' \to X' \times Y'$$ is marked anodyne.
In the situation of Proposition 3.1.2.1, to show that $p^{K^{\flat}} : (X^{\natural})^{K^{\flat}} \to (S^{\sharp})^{K^{\flat}} = (S^K)^{\sharp}$ has the right lifting property with respect to all marked anodyne morphisms, we have to show that every commutative diagram
\begin{CD}
    A @>a>> (X^{\natural})^{K^{\flat}}\\
    @V i V V @VV p^{K^{\flat}} V\\
    B @>>b> (S^{\sharp})^{K^{\flat}},
\end{CD}
where $i : A \to B$ is marked anodyne, admits a diagonal filler $h : B \to (X^{\natural})^{K^{\flat}}$. As you speculated in your question, the diagram above is equivalent to the diagram
\begin{CD}
    A \times K^{\flat} @>a'>> X^{\natural}\\
    @V i \times K^{\flat} V V @VV pV\\
    B \times K^{\flat} @>>b'> S^{\sharp},
\end{CD}
which is also commutative. For this latter diagram, Proposition 3.1.2.3 immediately implies the existence of a diagonal filler $h' : B \times K^{\flat} \to X^{\natural}$, whose transpose then is a diagonal filler for the first diagram.
