= Solution
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 maps
$$
\boxed{J=F(j)\varphi_0,\qquad M=F(m)\varphi_{A,A},\qquad E_A=\psi_0F(\varepsilon).}
$$
Its associativity follows from associativity of $m$ and the <lax monoidal functor> axiom for $\varphi$: 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:
$$
E_AM=\psi_0F(\varepsilon)F(m)\varphi_{A,A}=\psi_0F(\varepsilon m)\varphi_{A,A}.
$$
If $n:I\to A\otimes A$ is the original <coevaluation morphism>, the corresponding new one is $\psi_{A,A}F(n)\varphi_0$. The preceding <Frobenius monoidal functor> calculation proves both <snake identities>. Hence \b[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.
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