A monoidal category is a category equipped with a tensor bifunctor , a unit object , and coherence natural isomorphisms. The associator and the left and right unitors, are subject to Mac Lane’s pentagon and triangle coherence axioms, which force every bracketing to be canonically equal. It is braided when a natural satisfies the hexagons, and symmetric when moreover . The native syntax is the string-diagram calculus, where objects are wires, morphisms are boxes, and is vertical juxtaposition. Prototypical examples are and , and any category with finite products is cartesian monoidal. A one-object monoidal category is a monoid, whereas a monoidal category is itself a one-object bicategory, giving its slot in the periodic table of $n$-categories. A markov-category is a symmetric monoidal category in which copy and discard are supplied, and an enriched-category has its hom-objects living in one.
References
Saunders Mac Lane, “Natural associativity and commutativity”, Rice Univ. Studies 49 (1963)
André Joyal & Ross Street, “The geometry of tensor calculus I”, Adv. Math. 88 (1991)