I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated from maps in $S$ via directed colimits. This means every $f\in C$ can be written as a directed colimit of $f_i \in S$, and because $M$ is (co)complete this means the domain of $f$ and is the colimit of the domains of the $f_i$ and similarly for the codomain of $f$.

Is it true that $f$ is in $S$-cell (i.e. it's a retract of some transfinite compositions of pushouts of maps in $S$)? More generally, do we have $S$-cell $=$ $C$-cell?

A word of caution. Various facts about model categories will tell you that $S$-cell is closed under directed colimits and maybe even filtered colimits. But it seems to me that's for colimits taken in $M$ not in $Arr(M)$, i.e. it's saying that if we have a chain $X_0\to X_1\to \dots$ and each map $X_i \to X_{i+1}$ is in $S$-cell then the colimit $X_0\to X_\lambda$ is in $S$-cell. This is not what I'm asking for. I need a ladder $f_0\to f_1\to \dots$ and to know the colimit $f$ is in the same class $S$-cell which all the elements of the directed system are in. I'd like this for any directed colimit of the $f_i$ but I'm willing to accept it for a colimit of the sort just drawn, i.e. a sequential colimit. At least then I'd have an intuition for whether or not this is true.