In category theory, a **conservative functor** is a type of functor between two categories that preserves certain properties of objects and morphisms. Specifically, a functor \( F: \mathcal{C} \to \mathcal{D} \) is called conservative if it satisfies the following condition: A morphism \( f: A \to B \) in category \( \mathcal{C} \) is an isomorphism (i.e.
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A functor is conservative when it reflects isomorphisms: if is invertible then is invertible. It need not be full or faithful functor. The crude monadicity theorem uses this reflection property to lift invertibility of the comparison-adjunction counit from the base category.