X ↦ (X×A)^A

We can think of A as a machine's set of states; then (X × A)^A consists of all state machines on X with output to X, that is, all functions A → (A × X), the first component being transition, and the second an output to X.

Why is it a monad? u_X : X → (A × X)^A maps any element x ∊ X to a function that is identity on A and the constant x on X.
Let me express it in Lisp:

(define (ux x)
   (lambda (a) (list a x)))

How about m_X : (A × (A × X)^A)^A → (A × X)^A?

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