> there is nothing in the general monad interface that even allows you to go `m a -> a`, so something being a monad doesn't seem sufficient to be able to define

> there is nothing in the general monad interface that even allows you to go `m a -> a`, so something being a monad doesn't seem sufficient to be able to define that algebra. You've got the direction backwards: the point is to characterize all `m`-algebras, not to construct particular ones. This is closely related to the usual duality between types and values. It's very common for people new to TypeScript, for instance, to think that `{a: string} | {b: number}` should be `{a: string, b: number}`, because they're thinking primarily in terms of records as sets of fields, not types as sets of values. But making the type bigger makes the set of things you can say about its values (i.e. the space of functions out of it) smaller. Algebras are similarly the "value-like" counterparts to "space-like" monads.