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A Quantum Groups Primer, 1st Edition by Shahn Majid

B is a bialgebra map and the coaction of B is induced from flu by composing with it. By a bialgebra map in this proof, we mean of course a map between bialgebras respecting both the algebra and coalgebra structures. Meanwhile, the term 'push-out' is the arrow-reversal of the usual notion of 'pull-back' of modules under an algebra map: if it : C - D is a coalgebra map between two coalgebras, and if C coacts on something by a coaction 13, then (id ®7r) o,3 is a coaction of D on the same object, called the push-out coaction.

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