F-algebras - Category theory

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TheMatten

Can (B, f) where f : A -> B for some f, A, B be safely considered an F-Algebra for Const A?

TheMatten

Is it good idea to think of F-Algebra morphisms as morphisms that go from object with "more structure" to object with "same/less structure"? As in, their laws couldn't hold if I went from "less" to "more structure", could they?