I am working through the book "Frobenius Algebras and 2D TQFTs" and am stuck on an exercise: Show that the Frobenius relations and the (co)unit relations imply the (co)associativity relations.
It's obvious topologically, but I have tried manipulating the diagrams and can't find a solution using only these relations.
For examples of the diagrams involved see this other paper, Lemmas 3.15, 3.16 and 3.18.
Denote the product and co-unit by
and 
respectively (my diagrams are from top to down unlike the ones in your reference). One of the Frobenius relations is
Attaching
to the lower right leg, using the equalities
and
(where applications of the Frobenius law are marked in orange color) and applying the co-unit law we arrive at the desired associativity relation. A proof for co-associativity using the unit and co-product can be obtained by vertically reflecting the diagrams.