question about lambda-filtered colimits

[Gaucher Philippe <gaucher@pps.jussieu.fr>]

Dear category theorists


I would be interested in knowing a proof of the following fact (due to J.
Smith):

"In a combinatorial model category M (i.e. a locally presentable cofibrantly
generated model category), there are functorial factorizations of a map into
a trivial cofibration followed by a fibration which preserve lambda-filtered
colimits for sufficiently large regular cardinals lambda. The same is true
for the factorizations as a cofibration followed by a trivial fibration."

As far as I know about the proof, it suffices to apply the small object
argument step-by-step and then to use some property of lambda-filtered
colimits. The only property I know close to the problem is that a
lambda-filtered colimits of lambda-presentable objects is lambda-presentable.
But the underlying diagram of a pushout is not lambda-filtered. So I dont
understand...

Thanks in advance. pg.