F(\lim A) \simeq \lim (F\circ A) \,.
The archetypical example is the Set-valued hom-functor: its continuity in both arguments is indeed equivalent to the very definition of limit: for a diagram and , the covariant hom-functor satisfies by definition of limit
Hom_C(c, \lim F) \simeq \lim Hom(c,F(-)) \,.
Topologists sometimes use “continuous functor” to mean a functor enriched over Top, since a functor between topologically enriched categories is enriched iff its actions on hom-spaces are continuous functions.
Sheaf-theorists sometimes say “continuous functor” for a cover-preserving functor between sites, with the intuition being that it generalizes the inverse image induced by a continuous function of topological spaces.
H. Bass in his treatment of K-theory uses the older term ‘right continuous functor’ for the dual notion of cocontinuous functor in a version which is additive. If the domain of an additive functor which commutes with direct sums is a cocomplete category, then the functor automatically has right adjoint. Following this fact, some people in ring theory and noncommutative geometry use the simple term ‘continuous functor’ for a functor with a right adjoint (even if the domain abelian category is not cocomplete). In general, of course, this is just a bit more than cocontinuous in the standard sense.
Left I could understand, but right? —Toby
The way I rewrote it explains it. It is unfortunate that the Eilenberg-Watts theorem treated in Bass was using only right adjoint functors so later they dropped word right. – Zoran