The category of positive unital linear maps between C*-algebras is the Kleisli category of a comonad on the subcategory of unital *-homomorphisms between C*-algebras.

**Abstract**
In computer science the Kleisli category is often used to model
computation with a twist such as nondeterminism or probabilism.

Recently, Furber and Jacobs have shown in their study of quantum computation that the category of commutative C*-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C*-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit).

In this paper, we prove a non-commutative variant of this result: the category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the category of C*-algebras and MIU-maps