The notion of Kleisli 2-category over a 2-monad is the generalization of the notion of Kleisli category over a monad from category theory to 2-category theory (or from 2- to 3-category theory, depending on perspective).
Section 2.1.2 of
Ignacio Lopez Franco, Autonomous Pseudomonoids (pdf)
Susan Niefield, Exponentiability in homotopy slices of Top and pseudo-slices of Cat (TAC)
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