  We define a Frobenius algebra over fusion categories of the form
Rep$(G)\boxtimes$Rep$(G)$ which generalizes the diagonal subgroup of $G\times
G$. This allows us to extend field theoretical constructions which depend on
the existence of a diagonal subgroup to non-invertible symmetries. We give
explicit calculations for theories with Rep$(S_3)\boxtimes$Rep$(S_3)$ symmetry,
applying the results to gauging topological quantum field theories which carry
this non-invertible symmetry. Along the way, we also discuss how Morita
equivalence is implemented for algebras in symmetry categories.
