Identity types and weak factorization systems in Cauchy complete categories

It has been known that categorical interpretations of dependent type theory with ς- and Id-types induce weak factorization systems. When one has a weak factorization system on a category in hand, it is then natural to ask whether or not harbors an interpretation of dependent type theory with ς- and Id- (and possibly Π-) types. Using the framework of display map categories to phrase this question more precisely, one would ask whether or not there exists a class of morphisms of such that the retract closure of is the class and the pair forms a display map category modeling ς- and Id- (and possibly Π-) types. In this paper, we show, with the hypothesis that is Cauchy complete, that there exists such a class if and only if itself forms a display map category modeling ς- and Id- (and possibly Π-) types. Thus, we reduce the search space of our original question from a potentially proper class to a singleton.