nLab
cocontinuous functor

A functor is cocontinuous if it preserves small colimits.

Note that $F:C\to D$ is coconintinuous if and only if the functor $F^{\mathrm{op}}:C^{\mathrm{op}}\to D^{\mathrm{op}}$ 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 $F:{\mathrm{Set}}_{*}\to \mathrm{Set}$ from the category of pointed sets to the category of sets.

In general, 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.