The most general Thomas-Yau uniqueness, which includes immersed Lagrangians, seems to be due to Imagi [39]. The original Thomas-Yau theorem is phrased in terms of uniqueness in the Hamiltonian isotopy class, even though it can be cast in more general categorical terms. The categorical perspective is preferred, because it is closer to the spirit of homological mirror symmetry, and because one derived Fukaya category class may contain several Hamiltonian isotopy classes. If immersed Lagrangians are allowed, then isomorphism in $D^{b}Fuk(X)$ would also identify certain embedded Lagrangian objects with immersed objects of different topologies. When this happens, an interesting corollary of Thomas-Yau uniqueness is that at most one of these Hamiltonian classes contains special Lagrangian branes.