Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces.

Is it possible to treat the circles as ordered circles? To be specific, if a surface $S$ represents a morphism from $k$ circles to $m$ circles, starting with an order of $k$ circles can we induce an order on the $m$ circles so that it is well defined i.e respects the composition order.

As we know any $1+1$ dimensional TQFT corresponds to a Frobenius extension over a commutative ring. Is it true for the ordered cobordism category as well?