Dear List, Has the following elementary problem been already studied? Let C be a category, I a small category, F : I -> C a functor and alpha : F => c a cocone (c is an object of C). When does there exist a category D and a faithful functor G : C -> D taking alpha to a universal cocone? For example, if I is the empty category, the question becomes "when can you make c an initial object in a faithful way?". If I is the final category, then the cocone alpha amounts to a morphism f : F(*) -> c and the question becomes "when can you make f an isomorphism in a faithful way?". There are two obvious necessary conditions. 1) Let f,g : c -> d be two morphisms of C. If f alpha_i = g alpha_i for every i in I, then we should have f = g. 2) Let f : a -> F(i) and g : a -> F(j) be two morphisms of C such that alpha_i f = alpha_j g. Then for every cocone beta : F => d, we should have beta_i f = beta_j g. In the case of the empty category, the first condition means that for every object d there is at most one arrow c -> d and the second condition is void. In the case of the final category, the first condition means that f is an epi and the second that f is a mono. It is not hard to prove that in both cases, theses conditions are sufficient. Question: are they sufficient in the general case? Regards, -- Dimitri [For admin and other information see: http://www.mta.ca/~cat-dist/ ]