A C-system defined by a universe category

This is the third paper in a series. In it we construct a C-system CC(C,p) starting from a category C together with a morphism $p:\tilde{U} \to U$, a choice of pull-back squares based on $p$ for all morphisms to $U$ and a choice of a final object of C. Such a quadruple is called a universe category. We then define universe category functors and construct homomorphisms of C-systems CC(C,p) defined by universe category functors. In the sections before the last section we give, for any C-system CC, three different constructions of pairs ((C,p),H) where (C,p) is a universe category and $H : CC \to CC(C,p)$ is an isomorphism. In the last section we construct for any (set) category C with a choice of a final object and fiber products a C-system and an equivalence between C and the precategory underlying CC.