In mathematics, a monoidal category is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \to \mathbf {C} } that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism. Wikipedia

Category admitting tensor products Wikidata

A category C with a bifunctor ⊗ : C × C → C which may be called tensor product, an associativity isomorphism α A , B , C : ( A ⊗ B ) ⊗ C ≃ A ⊗ ( B ⊗ C ) , an object I which may be called tensor unit, a left unit natural isomorphism λ A : I ⊗ A ≃ A , a right unit natural isomorphism ρ A : A ⊗ I ≃ A , and some "coherence conditions" (pentagon and triangle commutative diagrams for those isomorphisms). Wiktionary