# instances of Fox's theorem

**Fox’s theorem** comes in multiple
versions: two of them describe the construction of the cofree and the free
cartesian category over a monoidal category. The more compact formulation says that any
symmetric monoidal category is a
cartesian category exactly if each object is a cocommutative comonoid and each morphism is a comonoid homomorphism, in a natural and uniform way.

The adjunctions of Fox’s theorem are hom-tensor adjunctions between the tensor of monoidal categories and the exponential of symmetric monoidal categories. These explain the requirements for naturality and uniformity in Fox’s theorem.

**References**

**Instances of Fox’s theorem.**

See: Fox’s theorem.