Definition:Continuous Functor

Definition
Let $\mathbf C$, $\mathbf D$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ be a functor.

Then $F$ is continuous iff for all diagrams $D: \mathbf J \to \mathbf C$ with limit ${\varprojlim \,}_j \, D_j$:


 * $F \left({{\varprojlim \,}_j \, D_j}\right) \cong {\varprojlim \,}_j \, F D_j$

where $F D: \mathbf J \to \mathbf D$ is the diagram obtained by composition of $F$ with $D$, and $\mathbf J$ is an arbitrary metacategory.

Also see

 * Functor Preserving Limits
 * Cocontinuous Functor