Just like a right -module can be tensored (over ) with a left -module to obtain an abelian group, a functor can be tensored (over ) with a functor to obtain an object of . Here, can be an arbitrary monoidal category. A simple case is with the cartesian product as monoidal structure.
Let be a category. Let be a monoidal category. Let and be functors. Then their tensor product is defined (if it exists) as the coend
The following slight variation is also important. Let be a category. Let be a category with all coproducts, so that the copower exists for any set and any object . Let and be functors. Then their tensor product is (if it exists) the coend
Intuition for the tensor product of functors can be gained by relating it to the tensor product of modules (see the first example below) and by a picture involving gluing specifications (see below).
Recall that a ring can be considered as an Ab-enriched category and that a right and a left module gives rise to an additive functor respectively . Then their tensor product as functors, calculated in the Ab-enriched setting and using the ordinary tensor product of abelian groups as monoidal structure on , coincides with their usual tensor product.
Let be a small category and let be a functor into a cocomplete category . Since the category of presheaves on is the free cocompletion of , the functor induces a functor . This functor can be explicitly described as .
Let denote the Yoneda embedding. Let be a presheaf on . Then . This fact is the co-Yoneda lemma (also referred to as the ninja Yoneda lemma in some circles).
From this perspective, the representable functor looks like a delta distribution concentrated at .
Intuition using gluing specifications
Recall that a presheaf can be seen as “gluing specification”: If is some functor into a cocomplete category , this gluing specification can be realized as . This colimit can also be written as the coend
that is as the tensor product . The tensor product can therefore be pictured as the ‘s, glued as specified by .
Since the Yoneda embedding includes an object into the category of formal gluing specifications, gluing the ‘s as specified by simply yields in ; thus .
Let be specifically the simplex category . Then is just a simplicial set, so the intuition of as a gluing specification is even more vivid. Tensoring with realizes this specification in the category of topological spaces, using standard -simplices as building blocks, and therefore yields the geometric realization .