# Definition:Functor Preserving Limits

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## Definition

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 ${\varprojlim \,}_j \, D_j$:

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

where $F D: \mathbf J \to \mathbf D$ is the composition of $F$ with $D$.