A comonad on $\mathrm{𝒞︀}$ is a monad $𝑊$ on ${\mathrm{𝒞︀}}^{\text{op}}$. Everything about its coalgebras is then inherited from the Eilenberg–Moore construction, instantiated at the opposite category — nothing is defined twice.

Algebras of $𝑊$ over ${\mathrm{𝒞︀}}^{\text{op}}$ are coalgebras $𝛾\in \mathrm{𝒞︀}(𝑥,𝑊𝑥)$ of the underlying endofunctor, and the monad algebra laws, read in ${\mathrm{𝒞︀}}^{\text{op}}$, are the comonad coalgebra laws — the unit and multiplication of $𝑊$, viewed in $\mathrm{𝒞︀}$, are the counit $𝜀$ and comultiplication $𝛿$:

The co-Eilenberg–Moore category is the opposite of the total category: