Free monads and their algebras

Today we are interested in the following theorem: The category of algebras of an endofunctor is isomorphic to the category of algebras of its free monad. It sounds complicated (and is rather not precise), so let me explain:

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Are idiom morphisms monad morphisms?

During the last Algebra of Programming meeting we were talking about idioms (applicative functors), monads, traversals, and such. At one moment a definition of idiom morphism appeared on the whiteboard, which is a function (a brave person might even say `natural transformation’) of type (for applicative M and N)

f :: M a -> N a

which respects the following:

f . pure = pure
f (mf <*> mx) = f mf <*> f mx

I was wondering: Sometimes it is the case that homomorphisms of simpler algebraic structures (for example, monoids) are authomatically homomorphisms of more complicated structures (for example, groups).

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