productpreserving
Product-preserving is a term used to describe a functor or, less commonly, a morphism, that preserves product structures under mapping. In category theory, a functor F: C → D is product-preserving if for all objects A and B in C there is a natural isomorphism F(A ×C B) ≅ F(A) ×D F(B), and F sends the terminal object 1C to a terminal object isomorphic to 1D. If the source category C has finite products, this condition means F preserves all finite products.
Equivalently, a functor is product-preserving when it preserves binary products and the terminal object, and the
Examples include forgetful functors from algebraic categories to Sets, which commonly preserve products. For instance, the
Not all functors are product-preserving; for example, the power-set functor P on Sets is not product-preserving
Related notions include finite-limit preservation and the role of product-preserving functors in transferring product structures across