Definition 5.5 (Kleisli category). Let $𝕋$ be a monad on $C$. The Kleisli category $C_{𝕋}$ is defined by $\mathrm{ob}{C}_{𝕋}=\mathrm{ob}C$, morphsims $A\stackrel{f}{\to }B$ in $C_{𝕋}$ are morphisms $A\stackrel{f}{\to }TB$ in $C$. The identity $A\stackrel{}{\to }A$ is $A\stackrel{{\eta }_A}{\to }TA$, and the composite of $A\stackrel{f}{\to }B\stackrel{g}{\to }C$ is $A\stackrel{f}{\to }TB\stackrel{Tg}{\to }TTC\stackrel{{\mu }_C}{\to }TC$.

For the unit and associative laws, consider the diagrams