There is a 2-category J-Colim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2-functor from J-Colim to the 2-category Cat of small categories is known to be monadic. We extend this result by considering not just conical colimits, but general weighted colimits; not just ordinary categories but enriched ones; and not just small classes of colimits but large ones; in this last case we are forced to move from the 2-category V-Cat of small V-categories to V-categories with object-set in some larger universe. In each case, the functors preserving the colimits in the usual ``up-to-isomorphism'' sense are recovered as the pseudomorphisms between algebras for the 2-monad in question.
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