= Initiality of the Kleisli adjunction
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.
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