\begin{definition}[Definition:Functor Preserving Limits]
Let $\mathbf C$, $\mathbf D$ and $\mathbf J$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be a functor.
Then $F$ '''preserves limits of type $\mathbf J$''' {{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 composition of $F$ with $D$.
\end{definition}