$T_1$ 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.