in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra. Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any (small) strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product). Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product. Every monoidal category can be seen as the category B(∗, ∗) of a bicategory B with only one object, denoted ∗. The concept of a category C enriched in a monoidal category M replaces the notion of a set of morphisms between pairs of objects in C with the notion of an M-object of morphisms between every two objects in C. Free strict monoidal category For every category C, the free strict monoidal category Σ(C) can be constructed as follows: its objects are lists (finite sequences) A1, ..., An of objects of C; there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m = n, and then the arrows are lists (finite sequences) of arrows f1: A1 → B1, ..., fn: An → Bn of C; the tensor product of two objects A1, ..., An and B1, ..., Bm is the concatenation A1, ..., An, B1, ..., B''m of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists. The identity object is the empty list. This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat. Specializations If, in a monoidal category, and are naturally isomorphic in a manner compatible with the coherence conditions, we speak of a braided monoidal category. If, moreover, this natural isomorphism is its own inverse, we have a symmetric monoidal category. A closed monoidal category is a monoidal category where the functor has a right adjoint, which is called the "internal Hom-functor" . Examples include cartesian closed categories such as Set, the category of sets, and compact closed categories such as FdVect, the category of finite-dimensional vector spaces. Autonomous categories (or compact closed categories or rigid categories) are monoidal categories in which duals with nice properties exist; they abstract the idea of FdVect. Dagger symmetric monoidal categories,