Kleisli Category

The Kleisli category of a Monad $T:\mathcal{C}\to\mathcal{C}$ is a Category $\mathcal{C}_{T}$ where

• Objects are objects of $\mathcal{C}$
• Morphisms ${{X}\to{Y}}$ are morphisms ${{X}\to{T(Y)}}$
• The identity morphism ${{X}\to{T(X)}}$ is given by the unit of the monad.
• Composition $f\circ g$ is given by $\mu\circ T(f)\circ g$

Equivalently, the Kleisli category is the Full Subcategory of the Eilenberg-Moore Category spanned by the Free Monad Algebras.