Completeness of cocompletions

We study several possible weakenings of the notion of limit and the associated notions of completeness for a category. We examine the relations among the various proposed notions of weakened completeness conditions. We use these conditions in the analysis of the existence of limits inside completions of categories under colimits. We further characterize when the completion of a category under finite colimits has finite limits with the aid of a condition requiring that reflexive symmetric graphs have bounded transitive hulls.