• Comonad
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$\gdef\ccal{\mathcal{C}}$ $\gdef\dcal{\mathcal{D}}$ $\gdef\lra{\leftrightarrow}$ $\gdef\Om{\Omega}$

What a comonad? Given two categories $$ L: \ccal \lra \dcal: R$$ We can form the comonad $\Om = LR \in End(\dcal)$, which have $$ \epsilon: \Om \to 1_\dcal, \quad L \eta R: \Om \to \Om \circ \Om $$