  In a recent paper we introduced a much weaker and easy to verify structure
than a model category, which we called a "weak fibration category". We further
showed that a small weak fibration category can be "completed" into a full
model category structure on its pro-category, provided the pro-category
satisfies a certain two out of three property. In the present paper we give
sufficient intrinsic conditions on a weak fibration category for this two out
of three property to hold. We apply these results to prove theorems giving
sufficient conditions for the finite accessibility of the category of weak
equivalences in combinatorial model categories. We apply these theorems to the
standard model structure on the category of simplicial sets, and deduce that
its class of weak equivalences is finitely accessible. The same result on
simplicial sets was recently proved also by Raptis and Rosick\'y, using
different methods.
