Reference

A continuous functor is one that preserves all small limits. One can make analogous definitions for colimits. For instance, a functor G preserves the colimits of F if G(L, φ) is a colimit of GF whenever (L, φ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.

A functor that preserves all small limits in C C that exist is called a continuous functor. Usually this term is only used when C C has all small limits, i.e. is a complete category. Examples. limits preserve limits. hom-functors preserve limits. adjoints preserve (co-)limits. the Yoneda embedding preserves limits (see there) Preservation of ...

In ordinary (Set -based) category theory, a functor F: C → D F \colon C \to D is continuous if it preserves all small limits that exist in its domain C C, i.e. given any small category diagram A: E → C A \colon E \to C in C C for which lim A \lim A exists, with universal cone lim A → A \lim A \to A, application of F F to that cone results ...