Understanding the small object argument (Q5896982)

Let \(J\) be a set of morphisms in a category \({\mathcal C}\). The small object argument is a transfinite construction used to obtain a weak factorisation system in \({\mathcal C}\). In this paper the author gives a modification of the small object argument with better properties than the standard argument: the transfinite sequences involved always converge; the factorisations are freely generated by \(J\) in a suitable sense; the process can be regarded as a special case of a well-known free monoid construction.