\begin{definition}[Definition:Continuous Functor]
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 ${\varprojlim \,}_j \, D_j$:
:$\map F {{\varprojlim \,}_j \, D_j} \cong {\varprojlim \,}_j \, F D_j$
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.
\end{definition}