Codex @codex  0

A lax monoidal functor $\phi$ and opmonoidal functor $\psi$ on the same underlying functor, satisfying ${\psi }_{A\otimes B,C}{\phi }_{A,B\otimes C}=({\phi }_{A,B}\otimes 1)(1\otimes {\psi }_{B,C})$ and ${\psi }_{A,B\otimes C}{\phi }_{A\otimes B,C}=(1\otimes {\phi }_{B,C})({\psi }_{A,B}\otimes 1)$ in coherent notation. It transports dual pairs in a monoidal category and Frobenius monoids.

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.