CLOSED MONOIDAL CATEGORY

In mathematics, a 'closed monoidal category' 'C' is a closed category with an associative tensor product (up to natural isomorphism) which is left adjoint to the internal Hom functor, that is a monoidal category equipped with a functor $Rightarrow$ such that the functor $Bmapsto(ARightarrow\; B)$ is right adjoint to the functor $Bmapsto(Aotimes\; B)$. This means that there exists a bijection between the Hom-sets
:$mathbf\{C\}(Aotimes\; B,\; C)congmathbf\{C\}(B,ARightarrow\; C)$
natural in ''B'' and ''C''.
Equivalently, a closed monoidal category 'C' is a category equipped, for every two objects ''A'' and ''B'', with

★ an object $ARightarrow\; B$,

★ a morphism $mathrm\{eval\}\_\{A,B\}\; :\; Aotimes\; (ARightarrow\; B)\; o\; B$,
satisfying the following universal property: for every morphism
:$f\; :\; Aotimes\; X\; o\; B$
there exists a unique morphism
:$h\; :\; X\; o\; ARightarrow\; B$
such that
:$f\; =\; mathrm\{eval\}\_\{A,B\}circ(mathrm\{id\}\_Aotimes\; h)$.
In particular, every cartesian closed category is a closed monoidal category.