Tensor products of finitely presented functors

We provide explicit constructions for various ingredients of right exact monoidal structures on the category finitely presented functors. As our main tool, we prove a multilinear version universal property so-called Freyd categories, which in turn is used proof correctness constructions. Furthermore, compare construction with Day convolution arbitrary additive always yields closed structure all In contrast, functor categories are not necessarily closed. necessary criterion being that relies underlying having weak kernels and prointernal hom structure. Our results stated constructive way thus serve as unified approach implementation tensor products contexts.