A product functor is a specific type of functor that takes two categories and produces their product in a way that respects the structure of both categories. It essentially combines objects and morphisms from two categories into a new category where objects are pairs of objects and morphisms are pairs of morphisms, thus creating a categorical product. This concept connects deeply to the ideas of covariant and contravariant functors, as well as adjunctions and exponential objects, showcasing how structures can be built from simpler components.
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