Kerodon

In the proof of 1.5.6.9, one reads: "Moreover, since $T$ is weakly saturated, the class $S$ is also weakly saturated." This implication holds of course, but (unless I'm missing something here) it involves a relatively long verification with quite a bit of diagram chases (and it makes use of the fact that $-\times A$ preserves colimits, which is something special about cartesian closed categories that doesn't hold in general). It's certainly more involved than the proof of 1.5.4.13 (to which you devoted the entire subsection 1.5.4). Perhaps it would be helpful to insert an exercise of the form "Let $K\rightarrow L$ be a morphism of simplicial sets and let $T$ be a weakly saturated class of morphisms of simplicial sets. Show that the class $S$ of morphisms $A\rightarrow B$ such that $B\times K \underset{A\times K}{⨿} A \times L \rightarrow B\times L$ belongs to $T$ is weakly saturated." (if not spelling out the details yourself).