Not every functor is cocontinuous; an example of a dis-cocontinuous (or disco-continuous) functor is the forgetful functor 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.
Revised on November 15, 2010 03:59:36
by Mike Shulman