coefficients in the $A$-bimodule $\tilde{A}$. If $G$ is cohomologically full and faithful, the left hand map is a quasi-isomorphism. Less obviously (it is a form of Morita invariance), if $G$ is a quasi-equivalence, the right hand map is also a quasi-isomorphism. Informally one can think of these two maps as follows. If we deform the $A_\infty$-structure on either $A$ or $\tilde{A}$ infinitesimally, there will be a term measuring the failure of $G$ to be an $A_\infty$-functor for the deformed structure, which is the image of the corresponding deformation classes in $\mathit{HH}(A,\tilde{A})$.

We also need to know about the behaviour under formal enlargement. Thinking of Hochschild cohomology in terms of deformation theory, one expects deformations of $A$ to induce ones of $A^{\mathit{tw}}$. This can indeed be made rigorous, leading to a canonical map $$\label{eq:twisted-gamma}
\Gamma^{\mathit{tw}}: \mathit{CC}(A,A) \longrightarrow \mathit{CC}(A^{\mathit{tw}},A^{\mathit{tw}}).$$ To see an example of how this looks like explicitly, take a Hochschild cochain $g$, and extend it in the obvious way to a cochain $g^{\oplus}$ on $A^{\oplus}$. Then the image $g^{\mathit{tw}} = \Gamma^{\mathit{tw}}(g)$ has leading term $$\label{eq:twisted-gamma-0}
g^{\mathit{tw},0} = g^{\oplus,0} + g^{\oplus,1}(\delta_X) + g^{\oplus,2}(\delta_X,\delta_X) + \cdots \in \mathit{hom}_{A^{\mathit{tw}}}^{|g|}(X,X).$$ Maybe more obviously, restriction to the full subcategory $A \subset A^{\mathit{tw}}$ yields a map 