Kleisli category (${\mathcal{C}}_T$)

For a monad $(T,\eta ,\mu )$ on $\mathcal{C}$, the Kleisli category ${\mathcal{C}}_T$ has the objects of $\mathcal{C}$ and morphismsIts identity on $A$ is ${\eta }_A$, and the composite of $f:A\to TB$ and $g:B\to TC$ is ${\mu }_{C}T(g)f:A\to TC$.

For a monad $(T,\eta ,\mu )$, the free functor $J:\mathcal{C}\to {\mathcal{C}}_T$ and functor $U:{\mathcal{C}}_T\to \mathcal{C}$ with $UA=TA$ and $U(f)={\mu }_{B}T(f)$ form an adjunction inducing that monad. Every other adjunction inducing the same monad receives the unique Kleisli comparison functor commuting with its left and right adjoints and preserving the adjunction structure. This is initiality among adjunctions inducing the specified monad, with morphisms required to respect that structure.

If $L⊣R$ induces a monad $T=RL$, the comparison maps $A$ to $LA$ and maps a Kleisli category arrow $f:A\to TB$ to ${\epsilon }_{LB}L(f)$. It is full and faithful, by the adjunction bijection $\mathcal{D}(LA,LB)\cong \mathcal{C}(A,TB)$. Consequently it is part of an equivalence of categories precisely when every object of $\mathcal{D}$ is isomorphic to some $LA$.

The free functor $F_T:\mathcal{C}\to {\mathcal{C}}_T$ is the identity on objects and sends $f:A\to B$ to ${\eta }_{B}f:A\to TB$. A functor $G:\mathcal{C}\to \mathcal{D}$ carries an algebra for the precomposition monad $G\mapsto GT$ exactly when it factors through $F_T$.