I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations of morphisms in $M$ do not extend to ones for $Pro-M$.
 My question is: which arguments and statements of Hovey's book cannot be applied in the absense of functorial factorizations? In particular, can one apply the dual to the argument used in the proof of Theorem 7.3.1 in Hovey? Though $Pro-M$ is not fibrantly generated, it seems that one can replace Hovey's (co)small object argument here with the one provided by Theorem 6.1 of B. Chorny's paper "A generalization of Quillen’s small object argument"; yet this is not quite clear from www.math.tamu.edu/~plfilho/wk-seminar/Hovey\_book.ps p. 186‎
(probably this version of Hovey's argument requires certain correction).


Also, is there an easier way to prove that $Ho(M)$ cogenerates $Ho(Pro-M)$?


Upd. I am deeply grateful to Tim Porter; it seems that the preprint <http://arxiv.org/abs/1305.4607> solves all my current problems with non-functorial factorizations. Yet comments concerning the general question could also be very interesting.



Which statements and arguments of Hovey's "Model categories" fail without functorial factorizations of morphisms?
Have a look at: Functorial Factorizations in Pro Categories by Ilan Barnea Tomer M. Schlank (ArXiv: <http://arxiv.org/abs/1305.4607>). This will in part answer your question. It is worth stating that Pro - M only fails to be fibrantly generated because there are too many generating fibrations, and Chorny has worked with another form of fibrant generation see his paper with Rosicky: Source: Homology Homotopy Appl. Volume 14, Number 1 (2012), 263-280.


