initial monad algebras are not fixpoints
It is not true that the initial monad algebra is always a fixpoint. For instance, the Maybe monad (• + 1) has the pointed spaces as algebras. The initial algebra is 1, but it is not a fixpoint: 1 + 1 ≠ 2.
This is a counterexample to Lambek’s lemma in the case of monad algebras.
What is indeed true is that the initial endofunctor algebra is always a fixpoint: for instance, the Maybe endofunctor has any pointed automorphism as an algebra. The initial pointed automorphism is the naturals with succ and zero, giving N + 1 ≅ N.
Back: algebra for a monad, Lambek’s lemma.