Let be a functor and a diagram. We say that creates limits for if has a limit whenever the composite has a limit, and both preserves and reflects limits of . This means that, in addition to having a limit whenever does, a cone over in is a limiting cone if and only if its image in is a limiting cone over .
Of course, a functor creates a colimit if creates the corresponding limit.
If creates all limits or colimits of a given type (i.e. over a given category ), we simply say that creates that sort of limit (e.g. creates products, creates equalizers, etc.).
A monadic functor creates all limits that exist in its codomain, and all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself). Creation of a particular sort of split coequalizer figures prominently in Beck’s monadicity theorem.
One should beware that in Categories Work, a more restrictive notion of “creation” is used which requires the every limit in to lift to one in uniquely on the nose, rather than merely up to isomorphism. This corresponds to a version of the monadicity theorem which asserts an isomorphism of categories, rather than merely an equivalence.