Is there a reasonable definition for a category of categories equipped with a factorization system, say $({\bf C},(E,M))$ and functors $F\colon ({\bf C},(E,M))\to ({\bf D}, (E', M'))$ that preserve them? I see several possible choices, all unreasonable in some sense:


1. $F(E)\subseteq E'$, and $F(M)\subseteq M'$; this is unreasonable, as it seems to strong $F=1\_{\bf C}$ does not send $(All, Iso)$ to $(Iso, All)$. Do I have to interpret this as an analogue of the identity being non-continuous if the domain topology is coarser?
2. $F$ preserves the left class. This is unreasonable, as it is not canonical. Why left, when $F$ is not a left adjoint?
3. $F$ preserves the right class. This is unreasonable, as it is not canonical. Why right, when $F$ is not a right adjoint?
4. $F(\mathbb F)\subseteq \mathbb{F}'$, if $\mathbb F =(E,M)$, and $\mathbb F' = (E', M')$. This means that $F(M)\subseteq M'$ and $F(E)\supseteq E'$.


I must admit I am confused.



On the category of categories with factorization system
The main theorem of Korostenski and Tholen's [Factorization systems as Eilenberg-Moore algebras](http://www.sciencedirect.com/science/article/pii/002240499390171O) proves that categories with (orthogonal) factorization systems are precisely the normal pseudo algebras for a 2-monad on Cat. The 2-monad in this case is the 2-monad that sends a category $C$ to its arrow category $C^2$ with the rest of the monad structure that you would guess (arising from the fact that the walking arrow category 2, like any category, is canonically a comonoid with respect to the cartesian product). 


How do we think about this result? To equip a category $C$ with the structure of an algebra is to define a functor $C^2 \to C$ that is a (strict) retraction of the diagonal functor $C \to C^2$. (This is the meaning of "normal" above.) Some clever diagram chasing proves that such a functor equips $C$ with a functorial factorization, with the functor $C^2 \to C$ sending a morphism to the object through which it factors. To say that this data defines a pseudo-algebra for the 2-monad demands an additional associativity condition, up to natural isomorphism (hence the "pseudo"), and this can be seen to imply that the left factor of a right factor is an isomorphism and that the right factor of a left factor is an isomorphism. These conditions, together with functoriality, suffice to show that the functorial factorization defines an orthogonal factorization system.


Note also that for any 2-monad $T$ on Cat, there is a related 2-monad $T'$ so that strict $T'$-algebras are exactly pseudo $T$-algebras.


So in particular, categories with a factorization system are also the strict algebras for a 2-monad, if you would prefer.


2-monad theory now provides a general framework within which to answer Fosco's question. In general, there are four natural kinds of morphisms for algebras for a 2-monad --- strict, pseudo, lax, and oplax --- with the strict ones (preserving everything on the nose) only rarely encountered in the wild. 


Suppose $C$ and $D$ are categories with factorization systems whose functorial factorizations are given by a pair of functors $Q : C^2 \to C$ and $E \colon D^2 \to D$ that send a morphism to the object through which it factors. A **lax morphism** of algebras specifies a functor $F \colon C \to D$ together with a natural transformation $EF \to FQ$ whose components define a commuting "comparison morphism" between the two factorizations of a map in the image of $F$. The presence of such a comparison implies that $F$ preserves the right classes. 


Dually, a **colax morphism** specifies a functor $F \colon C \to D$ together with a natural transformation $FQ \to EF$ the presence of which implies that $F$ preserves the left classes. By taking mates, one can see that if $F \dashv U$, then $F$ defines a colax morphism of algebras if and only if $U$ defines a lax morphism of algebras, which explains why lax morphism structure feels natural for right adjoints and colax morphism structure feels natural for left adjoints.


Some preliminary remarks along these lines in the more general setting of [algebraic weak factorization systems](https://ncatlab.org/nlab/show/algebraic+weak+factorization+system) can be found in my thesis or (in a much more satisfying account) in Bourke and Garner's [Algebraic weak factorization systems I](https://arxiv.org/abs/1412.6559).


