$T_1$ and $T_2$ are our dual theories.

Naive account of equivalence: $\exists$ map between theories that relate respective physical quantities.

Categorical equivalence is a somewhat meta-level version of isomorphism.  Only real way we have to talk about inter (mathematical) theory relations.

Sometimes the maps are the result of a happy coincidence.  Sometimes e.g. Legendre transformation we have a more meaningful example.  Links together two theoretical structures (in distinct theories) with a similar role.