distributed systems

let's invent a cool distributed "site" thing (not an actual site)

A site is given by a category C and a set Cov(C) of families of morphisms with fixed target {U_i -> U}_{i in I} satisfying the following axioms
(1) If V -> U is an isomorphism then {V -> U} in Cov(C).
(2) If {U_i -> U}_{i in I} in (C) and for each i we have {V_ij -> U_i}_{j in Ji} in Cov(C), then {V_ij -> U}_{i in I, j in Ji} in Cov(C).
(3) If {U_i -> U}_{i in I} in Cov(C) and V -> U is a morphism of C then U_i x_U V exists for all i and {U_i x_UV -> V}_{i in I} in Cov(C).