# Factorization systems on tensor product of presentable categories

This question is motivated by the following particular problem.

I have two presentable categories $\cal A,B$ with orthogonal factorization systems $({\cal E}, {\cal M})$ (on A) and $({\cal U},{\cal V})$ (on B). I want to produce a factorization system on the tensor product ${\cal A}\otimes{\cal B}$ out of these data.

This is the standard way, I think:

1. Consider the product of factorization systems on ${\cal A}\times{\cal B}$; this is precisely what you think.

2. Consider the image of the left class ${\cal E}\times{\cal U}$ under $\otimes$, call it $\Sigma = {\cal E}\otimes{\cal U}$

3. The (=a strongly orthogonal variant of the) small object argument used in ${\cal A}\otimes{\cal B}$ gives that the pair $$({\cal L},{\cal R}) = (\text{Sat }\Sigma, \Sigma^\perp)$$ (on the left, saturation: closure under retracts, transfinite compositions and pushout; on the right, the right orthogonal to $\Sigma$) is a orthogonal factorization system on ${\cal A}\otimes{\cal B}$.

Now, I'm interested in having some nice properties of $({\cal E}, {\cal M})$ and $({\cal U}, {\cal V})$ preserved by this construction; like for example if both $({\cal E}, {\cal M})$ and $({\cal U}, {\cal V})$ are proper, generated by small classes of small objects, reflective/coreflective (i.e. all the classes have the 2-out-of-3 property), semi-exact, simple, normal... then also $(\cal L,R)$ has these properties.