A functor $F : C \to D$ is continuous if it preserves all small (weighted) limits that exist in $C$, i.e. if for every small category diagram $A : E \to C$ in $C$ there is an isomorphism
Since all limits can be obtained from (small) products and binary equalizers, it follows that a functor is continuous if and only if it preserves all products and all binary equalizers.
The adjoint functor theorem states that any continuous functor between complete categories has a left adjoint if it satisfies a certain ‘small solution set’ criterion.
If $C$ has finite limits, then functors commuting with these finite limits are precisely what are called left exact functors. Sometimes they are called “finitely continuous.”
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
Thanks. —Toby