A C-system defined by a universe category

Abstract: This is a major update of the previous version. The methods of the paper are now fully constructive and the style is "formalization ready" with the emphasis on the possibility of formalization both in type theory and in constructive set theory without the axiom of choice. This is the third paper in a series started in 1406.7413. In it we construct a C-system $CC({\cal C},p)$ starting from a category $\cal C$ together with a morphism $p:\widetilde{U}\rightarrow U$, a choice of pull-back squares based on $p$ for all morphisms to $U$ and a choice of a final object of $\cal C$. Such a quadruple is called a universe category. We then define universe category functors and construct homomorphisms of C-systems $CC({\cal C},p)$ defined by universe category functors. As a corollary of this construction and its properties we show that the C-systems corresponding to different choices of pull-backs and final objects are constructively isomorphic. In the second part of the paper we provide for any C-system CC three constructions of pairs $(({\cal C},p),H)$ where $({\cal C},p)$ is a universe category and $H:CC\rightarrow CC({\cal C},p)$ is an isomorphism. In the third part we define, using the constructions of the previous parts, for any category $C$ with a final object and fiber products a C-system $CC(C)$ and an equivalence $(J^,J_):C \rightarrow CC$.