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

我们在有限呈现的函子类别上为右精确单曲面结构的各种成分提供了明确的构造。作为我们的主要工具，我们证明了所谓的 Freyd 范畴的普遍属性的多线性版本，这反过来又被用于证明我们的结构的正确性。此外，我们将我们的构造与任意加法函数的 Day 卷积进行比较。日卷积总是在所有的类别上产生一个封闭的幺半群结构加法函数。相反，有限呈现的函子类别的右精确幺半群结构不一定是封闭的。我们提供了一个封闭的必要标准，该标准依赖于具有弱核的基础类别和所谓的有限呈现的前内部 hom 结构。我们的结果以建设性的方式陈述，因此可以作为在各种情况下实现张量积的统一方法。