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](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf%22Notes%20on%20quasi-categories%22)"). 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.



Quasi-categorical factorization system induced on $X^S$
This will follow from the reflectivity of the right class $M$ in $X^I$ (the reflection just maps $f=me$ to its right half, $m$, which is essentially unique.) The reflection $X^I\to M$ extends to make $M^S$ reflective in $(X^S)^I$ for every $S$, so one has only to check that this is the right kind of reflective subcategory of the arrow category to give a factorization system on $X^S$. The "right kind" is precisely a reflective subcategory for which the unit $f\to m$ of the reflection has trivial bottom edge: then the left half of the factorization system comes as the top edge. You can see this explained for categories in 2.4 of Garner's [paper](http://comp.mq.edu.au/~rgarner/Papers/CT07.pdf "Garner") *Understanding the Small Object Argument*. Since everything in Joyal's definition really happens in the 2-category of quasicategories-specifically, the trivial spaces of solutions to lifting problems, the adjunctions, and the identity arrows-I'm pretty confident the same story goes through just fine here. 


I'm pretty sure this won't work with homotopy factorization systems on model categories. You're asking precisely that the factorizations be choosable functorially, but the classes in a homotopy factorization system don't have a 1-categorical universal property making this possible. I suppose this should be possible in most examples, if I'm correctly understanding what the usual examples are, such as (n-connected,n-truncated), or the (local trivial cofibration,local fibration.) For instance, the latter will always lift to diagram categories when the diagram categories admit compatible model structures, as they usually do.


