Leave $ C $ Be a small category, and consider the kind of diagrams. $ G: D a C $, with $ D $ A small category, which has colimits in. $ C $. This is an appropriate class even when $ C $ It is very small, eg. when $ D $ has a terminal object $ t $, any functor $ G: D a C $ has a colimit $ G (t) $, and there is an appropriate class of small categories with a terminal object.
However, these colimits feel somewhat "trivial"; in some cases, at least, we can find a small set of diagrams that "carry all the non-trivial information" on the colimit diagrams in $ C $. For example, yes $ C $ it is a poset, then it is enough to consider injective functors $ G $ (and we can also take $ D $ to be discrete too), and these form an essentially small set. For a non-posetal $ C $ we can not restrict ourselves to injective functors, since co-products are not idempotent, but there may be some other restriction that works. Note that according to Freyd's theorem, a small non-posetal category has a limit on the cardinality of the coproducts it can admit; but this does not answer the question itself, since a particular colimit can exist even if the coproducts that would be necessary to build it from co-products and co-factors do not.
Here are two ways to ask the precise question:
Given a small category $ C $, there is a small set $ L $ of diagrams $ G: D a C $ With colimits such that for any diagram. $ G & # 39 ;: D & # 39; a C $ with a colimit, there is a $ (D, G) in L $ and a final functor $ F: D to D & # 39; $ such that $ G = G & # 39; circ F $?
Given a small category $ C $, there is a small set $ L $ of diagrams $ G: D a C $ with colimits such that if a functor $ H: C a E $ Keep the colimits of all the diagrams in. $ L $, then it keeps all the colimits that exist in $ C $?
Any solution to question 1 is also a solution to question 2, but I'm not sure if the opposite is true. The mention of Freyd's previous theorem suggests that a solution might require a classical logic; I would find it more surprising if such a set existed for a small complete non-posetal category, although I do not immediately see an argument that I can not.
Of course, you can also ask similar questions for rich categories, internal categories, $ infty $-categories, and so on. Bonus points go to a response that applies more generally in such contexts.