Kleisli comparison functor
= Kleisli comparison functor
{c}
{title2=$K:\mathcal C_T\to\mathcal D$}
If $L\dashv R$ induces a <monad> $T=RL$, the comparison maps $A$ to $LA$ and maps a <Kleisli category> arrow $f:A\to TB$ to $\varepsilon_{LB}L(f)$. 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 $LA$.