monoidal category

Noun

 * 1)  A category $$\mathcal{C}$$ with a bifunctor $$\otimes: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$$ which may be called tensor product, an associativity isomorphism $$\alpha_{A,B,C} : (A \otimes B) \otimes C \simeq A \otimes (B \otimes C)$$, an object $$I$$ which may be called tensor unit, a left unit natural isomorphism $$\lambda_A : I \otimes A \simeq A$$, a right unit natural isomorphism $$\rho_A : A \otimes I \simeq A$$, and some "coherence conditions" (pentagon and triangle commutative diagrams for those isomorphisms).