coalgebra

A coalgebra is a vector space $A$ over a field $𝕂$ with a $𝕂$-linear map $\mathrm{\Delta }:A\to A\otimes A$,called the comultiplication,and a (non-zero) $𝕂$-linear map $\epsilon :A\to 𝕂$,called the counit, such that

$(\mathrm{\Delta }\otimes \mathrm{id})\circ \mathrm{\Delta }$$=$$(\mathrm{id}\otimes \mathrm{\Delta })\circ \mathrm{\Delta }\mathit{\hspace{1em}}\text{(coassociativity)},$

$(\epsilon \otimes \mathrm{id})\circ \mathrm{\Delta }$$=\mathrm{id}=$$(\mathrm{id}\otimes \epsilon )\circ \mathrm{\Delta }.$

In of commutative diagrams: