Kleisli category

Commutative diagram of function composition in a Kleisli category. Given a monad ${\displaystyle (T,\eta ,\mu )=(T:{\mathcal {C}}\rightarrow {\mathcal {C}},\ \eta$ :{\mbox{id}}_{\mathcal {C}}\rightarrow T,\ \mu :T^{2}\rightarrow T)} , consider a Kleisli category ${\displaystyle {\mathcal {K}}}$ over that monad. Morphisms ${\displaystyle {\tilde {f}}:A\rightarrow B}$, ${\displaystyle {\tilde {g}}:B\rightarrow C}$, and ${\displaystyle {\tilde {h}}={\tilde {g}}\circ {\tilde {f}}:A\rightarrow C}$ in ${\displaystyle {\mathcal {K}}}$ correspond to morphisms ${\displaystyle f:A\rightarrow TB}$, ${\displaystyle g:B\rightarrow TC}$, and ${\displaystyle h:A\rightarrow TC}$ in ${\displaystyle {\mathcal {C}}}$, respectively. The composition rule is ${\displaystyle {\tilde {g}}\circ {\tilde {f}}={(\mu _{{\text{cod}}(g)}\circ Tg\circ f)}^{\sim }}$. The Kleisli category shares the same objects as its underlying category. The morphisms of the Kleisli category (e.g.: ${\displaystyle {\tilde {f}}}$ and ${\displaystyle {\tilde {g}}}$) are embellished versions of the morphisms of its underlying category (e.g.: f and g), and they are derived from those of the underlying category by means of applying a monad to their codomains.