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A functor creates categorical limits of a specified shape when every given limiting categorical cone over an image diagram uniquely lifts to a categorical cone over the original diagram, and the lift is limiting. This is stronger than just being a limit-reflecting functor. Forgetting arrows from a functor category creates its pointwise limits in a functor category.

For a diagram $D:\mathcal{J}\to \mathcal{C}$, a limit-preserving functor carries every limiting cone over a diagram to a limiting categorical cone. A limit-reflecting functor has the converse property: a categorical cone is limiting whenever its image is limiting. A limit-creating functor uniquely lifts every specified limiting categorical cone over the image diagram to a categorical cone over $D$, and the lift is limiting. These definitions concern diagrams of the stipulated shape; creation includes the lifting requirement, not merely reflection.

Let $(L,(p_j))$ be a categorical limit of $D$. The mapis a bijection: the right side consists exactly of compatible families of arrows from $A$, and the categorical limit's universal property gives their unique factorization through $L$. The bijection is induced by the categorical limit projections, so it proves that covariant representables preserve limits, including the empty diagram and its terminal object.