nLab cocontinuous functor

A functor is cocontinuous if it preserves small colimits. Usually one only considers cocontinuous functors whose domain and codomain are cocomplete categories (have all small colimits).

Note that $F: C \to D$ is cocontinuous if and only if the functor $F^{op}: C^{op} \to D^{op}$ between opposite categories is a continuous functor.

Not every functor is cocontinuous; an example of a dis-cocontinuous (or disco-continuous) functor is the forgetful functor $F : Set_* \rightarrow Set$ from the category of pointed sets to the category of sets.

“Morally speaking,” a functor is cocontinuous if and only if it is a left adjoint (or equivalently has a right adjoint). Actually, only the ‘if’ part is true as stated; the ‘only if’ part has some conditions on it, given by the adjoint functor theorem.