TITLE: Big list of comonads
QUESTION [27 upvotes]: The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics. 
The dual concept, a comonad, is less popular. 
What are examples of comonads, in different categories, and different fields of math?

REPLY [3 votes]: The so-called game comonads have been recently studied in the context of finite model theory. The main references are the following:

Samson Abramsky, Anuj Dawar, and Pengming Wang. "The pebbling comonad in finite model theory." 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, 2017. (see also arXiv:1704.05124)
Samson Abramsky and Nihil Shah. "Relating Structure and Power: Comonadic semantics for computational resources." International Workshop on Coalgebraic Methods in Computer Science. Springer, Cham, 2018. (see also arXiv:1806.09031)

Game comonads are an active area of research. In nutshell, some of the well-known model comparison can be turned into a comonad. This is done by representing the states of the game semantically. For example, the  Ehrenfeucht-Fraïssé comonad $\mathbb E_k$ is a comonad on the category of relational structures. The universe of $\mathbb E_k(A)$ consists of non-empty words of length ${\leq}k$, where the alphabet is taken to be the universe of $A$. A word $[a_1,a_2,\dots,a_n]$ represents spoiler's moves in $A$ in the one-way Ehrenfeucht-Fraïssé game.