I am reading the lecture note by Căldăraru:https://arxiv.org/pdf/math/0501094, in the last chapter of this note, he said that we should consider dg category instead of the derived category of coherent sheaves to realize the slogan Hochschild cohomology is natural transformation.
He shows that for ellptic curve $E$, $\mathbf{HH}^2(E)\cong H^1(E,T_E)\cong H^0(E, \mathrm{O}_E)\cong\mathbb{C}$ whreas the natural transformation $\mathrm{Id}\to \mathrm{id[2]}$ is always zero, since for any coherent sheaves $F$ over $E$ the 2nd extension group $\mathrm{Ext}^2(F, F)=0$ and any object in $D^b_{Coh}(E)$ splits into its cohomology sheaves.
I wonder why the same argument fails to be applied to dg category? I know nothing about dg category but any explanation is welcome.
