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We show that the canonical equivalences of categories between 2-dimensional(unoriented) topological quantum field theories valued in a symmetric monoidalcategory and (extended) commutative Frobenius algebras in that symmetricmonoidal category are symmetric monoidal equivalences. As an application, werecover that the invariant of 2-dimensional manifolds given by the product of(extended) commutative Frobenius algebras in a symmetric tensor category is themultiplication of the invariants given by each of the algebras.

Symmetric monoidal equivalences of topological quantum field theories in dimension two and Frobenius algebras