An appropriate notion of morphism between finitely complete categories , is a left exact functor, or a functor that preserves finite limits (also called a lex functor, a cartesian functor, or a finitely continuous functor). A functor preserves finite limits if and only if:
it preserves terminal objects, binary products, and equalizers; or
it preserves terminal objects and binary pullbacks.
Since these conditions frequently come up individually, it may be worthwhile listing them separately:
preserves terminal objects if is terminal in whenever is terminal in ;
preserves binary products if the pair of maps
exhibits as a product of and , where and are the product projections in ;
preserves equalizers if the map
is the equalizer of , whenever is the equalizer of in .