Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a **<blank> cofibration** if for any cofibration $i:A\to B$ in $\cal M$, the "Leibniz" pushout corner map

$$ S(B) \cup_{S(A)} T(A) \longrightarrow T(B)$$

is a cofibration in $\cal N$ that is acyclic if $i$ is. Has this notion been studied? Does it have a name?

Some notes:

- If $H:\cal P\times \cal M\to \cal N$ is a Quillen two-variable adjunction (such as the tensor product of a monoidal model category, or the copower of an enriched model category), then $H(j,-) : H(C,-) \to H(D,-)$ is a <blank> cofibration for any cofibration $j:C\to D$ in $\cal P$.
- <blank> cofibrations are closed under pushout and transfinite composites in $[{\cal M, N}]$. In fact they are the maps in $[{\cal M,N}]$ having the left lifting property with respect to a certain class of "Leibniz right Kan extension" maps defined from a cofibration in $\cal M$ and a fibration in $\cal N$ one of which is acyclic. But I see no obvious reason for these classes of maps to form a weak factorization system.
- If $S$ (hence $T$) is <blank> cofibrant, then $\alpha:S\to T$ is a <blank> cofibration if and only if the induced functor $\cal M \to \cal N^{\bf 2}$ is left Quillen for the Reedy model structure on $\cal N^{\bf 2}$ where the arrow of $\bf 2$ points "up". I think the cofibrancy condition can be removed by using instead some "hybrid Reedy-ish" weak factorization systems on $\cal N^{\bf 2}$.