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