Describing where a Kleisli Triple fits into a Monad ontology

Question

I'm trying to map a Kleisli triple onto my existing understanding of Monads.

I can represent my understanding of Monads like this: (courtesy of Jim Duey's slides at 13)

Could you please point to the part on this diagram where Kleisli triples fit in - or even better - draw another diagram that this diagram can fit into that explains it?

Answer 1

Kleisli triples fit in the diagram exactly where you have "monads". Kleisli triples are equivalent to monads (you might say they are one presentation of monads).

I infer from the link that you are thinking about this in the context of programming languages? This question/answers might help: What is a monad in FP, in categorical terms?.