Quasi-Hopf and weak-Hopf extensions change different axioms

The two extensions alter different parts of the bialgebra axioms. A quasi-bialgebra keeps $\mathrm{\Delta }$ and $\epsilon$ unital algebra maps and replaces strict coassociativity by conjugation with $\mathrm{\Phi }$. A weak bialgebra keeps strict coassociativity and multiplicativity of $\mathrm{\Delta }$, but weakens the unit and counit identities; its module tensor unit is the target base $H_t$. Thus the reassociator and the source/target base data are distinct structures. They have a common special case: an ordinary Hopf algebra with bijective antipode is quasi-Hopf with $\mathrm{\Phi }=1$ and $\alpha =\beta =1$, and is also weak Hopf by the ordinary bialgebra and antipode identities. The ordinary antipode is an algebra antihomomorphism by Uniqueness and anti-multiplicativity and anti-comultiplicativity of the antipode, so it meets the quasi-antipode map requirement. Bijectivity is required by Quasi-bialgebras, quasi-antipodes, and quasi-Hopf algebras, while Hopf algebras and antipodes does not impose it. The companion examples exhibit a nontrivial associator with the ordinary unit and counit, and a nontrivial weak base with strict coassociativity.