Comonad

Remarks

A comonad on $\mathcal{C}$ is the same as a monad on the opposite category $\mathcal{C}^{\mathrm{op}}$.

Example: fix a set $A$ and let $G(X) = A \times X$ on the category of sets, with $G(f) = \id_A \times f$. Put $\varepsilon_X(a, x) = x$ and $\delta_X(a, x) = (a, (a, x))$. Both sides of the coassociativity law send $(a, x)$ to $(a, (a, (a, x)))$. The counit laws hold since $\varepsilon_{G(X)}(a, (a, x)) = (a, x)$ and $G(\varepsilon_X)(a, (a, x)) = (a, x)$.

Non-example: let $A$ have two elements $a_0 \ne a_1$, and keep $G$ and $\varepsilon$ but put $\delta_X(a, x) = (a_0, (a, x))$. Then $G(\varepsilon_X)(\delta_X(a_1, x)) = (a_0, x) \ne (a_1, x)$, so a counit law fails.

If $F \colon \mathcal{C} \to \mathcal{D}$ is left adjoint to $U \colon \mathcal{D} \to \mathcal{C}$ with counit $\varepsilon$ and unit $\eta$, then $(F U, \varepsilon, F \eta U)$ is a comonad on $\mathcal{D}$.