I have a question concerning Remark 3.1.1.3 in Lurie's "Higher Topos Theory" about the definition of the class of marked anodyne morphisms. There, it is mentioned that "it suffices to allow $K$ to range over a set of representatives for all isomorphism classes of Kan complexes with only countably many simplices". Perhaps I'm wrong, but wouldn't it even be possible to replace the class (4) by the class (4') containing only the inclusion $K^{\flat} \hookrightarrow K^{\sharp}$, where $K$ is the simplicial set having two 0-simplices ( - $x$ and $y$ - ), two 1-simplices ( - one going from $x$ to $y$ and the other going from $y$ to $x$ - ), and two 2-simplices witnessing that the two 1-simplices shall be equivalences, and besides that only degenerate simplices.
This is intimately related to the proof of Proposition 3.1.1.6, specifically line 7 on page 150. There, $K$ is defined to denote the largest Kan complex contained in $X_{p(x)}$; but wouldn't it also be possible to define $K$ to be the simplicial set described above, and, after choosing an extension of $d_2(\sigma) : \Delta^1 \to X$ to $K$, argue that $d_2(\sigma)$ must be marked because $p$ has the right lifting property with respect to the inclusion $K^{\flat} \hookrightarrow K^{\sharp}$?