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In terms of string diagrams, this definition says:
The first line here shows the associative law and left/right unit laws for a monoid. The second line shows the coassociative law and left/right counit laws for a comonoid. The third line shows the Frobenius laws.
Certain kinds of Frobenius algebras have nice PROPs or PROs. The PRO for Frobenius algebras is the monoidal category of planar thick tangles, as noted by Aaron Lauda Lauda (2006) and illustrated here:
… [A]ny commutative Frobenius algebra gives a 2d TQFT…. The monoid laws look like this:
The comonoid laws look like this:
The Frobenius laws look like this:
and the commutative law looks like this: