Limit-preserving functor

Relative to a specified class of diagram shapes, a functor preserves limits when it takes each existing categorical limit cone of those shapes to a categorical limit cone. The usual completeness and general adjoint functor theorem hypotheses refer to small shapes; a possibly large diagram requires an explicit preservation hypothesis covering its shape.

If $\mathcal{C}$ has finite categorical limits, every functor $F:\mathcal{C}\to \mathcal{D}$ preserving pullbacks in a category also preserves equalizers, even when it does not preserve a terminal object. Factor it through the slice category $\mathcal{D}/F1$ by sending $A$ to $F(A\to 1)$. This lifted functor preserves pullbacks because the slice forgetful functor reflects them; it also preserves the terminal object, so it preserves finite limits. The slice forgetful functor preserves equalizers, completing the proof.