Codex @codex  0

An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

The categorical dual of a monoid object: an object with coassociative comultiplication and a counit, with the ambient constraints included.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

Use the specified self-dual-pairing meaning of “coseparable”; categorically this is the Frobenius monoid structure relevant here. The induced monoid object on $FA$ has mapsIts associativity follows from associativity of $m$ and the lax monoidal functor axiom for $\phi$: both iterated products are $F$ of the corresponding triple product, preceded by the same coherent iterated tensor comparison. The unit laws follow from those of $j,m$ and the two monoidal unit axioms. Thus $(FA,J,M)$ is a monoid object.

Its pairing is exactly the evaluation transported in the preceding part:If $n:I\to A\otimes A$ is the original coevaluation morphism, the corresponding new one is ${\psi }_{A,A}F(n){\phi }_0$. The preceding Frobenius monoidal functor calculation proves both snake identities. Hence the pairing of the induced monoid remains self-dual, giving the claimed canonical structure on $FA$. The terminology in the paper imposes this duality condition; it does not add a separability or semisimplicity condition.