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 $\mathrm{𝒞︀}$. Write $𝜀:𝐹\circ 𝑈\Rightarrow \text{𝖨𝖽}$ for the counit of the adjunction. Every object $𝑑$ of $\mathrm{𝒟︀}$ then induces a $𝑇$-algebra carried by the object $𝑈𝑑$, witnessed by the map

This assignment extends to a functor into the Eilenberg–Moore category,

the comparison functor of the adjunction.

Dually, an adjunction induces a comonad on the other side and a comparison into the co-Eilenberg–Moore category. When these comparisons are equivalences we say that the adjunction $𝐹⊣𝑈$ is (co)monadic.