Let $X$ be an $\infty$-category (I am happy to assume it is bicomplete and stable, but this should not be necessary) and consider a factorization system $F=(E,M)$ on $X$ (this is defined in Section 24 of Joyal's "Notes on quasi-categories"). In 24.10, Joyal mentions without proof that, given any simplicial set S, there is a, pointwise induced, factorization system $F_S$ on $X^S$.
My questions are the following:
- Is there any easy argument to show that $F_S$ above is a factorization system?
- Is there any similar result for homotopy factorization systems in the setting of model categories? For this I mean, given a model category $(M,W,Fib,CoFib)$, a homotopy factorization system $F$ on $M$, and a small category $I$. If we define $F_I$ pointwise, do we obtain a homotopy factorization system of the relative category $(M^I,W_I)$?
- Of course I would be happy to know about related results in different contexts as, for example, DG-categories.