A C-system defined by a universe in a category

This is the third paper in a series started in 1406.7413. In it we construct, in a functorial way, a C-system $CC({\cal C},p)$ starting from a category $\cal C$ together with a morphism $p:\widetilde{U}\rightarrow U$ assuming that $\cal C$ has a final object and pull-backs of $p$.