An adjunction $𝐹⊣𝑈$ with $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒟︀}$ and $𝑈:\mathrm{𝒟︀}\to \mathrm{𝒞︀}$ induces a monad $𝑇=𝑈\circ 𝐹$ on $\mathrm{𝒞︀}$, and a comparison functor

sending each object of $\mathrm{𝒟︀}$ to the $𝑇$-algebra it carries.

The functor $𝑈$ is monadic when $𝐾$ is an equivalence: the adjunction exhibits $\mathrm{𝒟︀}$ as objects of $\mathrm{𝒞︀}$ equipped with algebraic structure for $𝑇$, the Eilenberg–Moore category.

Comonadicity is monadicity in the opposite category: a left adjoint $𝐿:\mathrm{𝒟︀}\to \mathrm{𝒞︀}$ with right adjoint $𝑅$ induces a comonad $𝑊=𝐿\circ 𝑅$ on $\mathrm{𝒞︀}$, a comparison $\mathrm{𝒟︀}\to \text{coEM}(𝑊)$ into the co-Eilenberg–Moore category, and $𝐿$ is comonadic when this comparison is an equivalence.