In mathematics, a closed monoidal category C is a closed category with an associative tensor product which is left adjoint to the internal Hom functor, that is a monoidal category equipped with a functor $\backslash Rightarrow$ such that the functor $B\backslash mapsto(A\backslash Rightarrow\; B)$ is right adjoint to the functor $B\backslash mapsto(A\backslash otimes\; B)$. This means that there exists a bijection between the Hom-sets

$\backslash mathbf\{C\}(A\backslash otimes\; B,\; C)\backslash cong\backslash mathbf\{C\}(B,A\backslash Rightarrow\; C)$

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 $A\backslash Rightarrow\; B$,
• a morphism $\backslash mathrm\{eval\}\_\{A,B\}\; :\; A\backslash otimes\; (A\backslash Rightarrow\; B)\backslash to\; B$,

satisfying the following universal property: for every morphism

$f\; :\; A\backslash otimes\; X\backslash to\; B$

there exists an unique morphism

$h\; :\; X\; \backslash to\; A\backslash Rightarrow\; B$

such that

$f\; =\; \backslash mathrm\{eval\}\_\{A,B\}\backslash circ(\backslash mathrm\{id\}\_A\backslash otimes\; h)$.

In particular, every cartesian closed category is a closed monoidal category.

Monoidal categories