in reverse direction to . These two maps are inverse quasi-isomorphisms (strict inverses in one order, and inverses up to homotopy in the other), which is one form of the derived invariance of Hochschild cohomology. There is also an analogue for $A$-modules. Again thinking in terms of deformations, a deformation of $A$ induces a deformation of $A^{\mathit{mod}}$, but one which remains within the class of dg categories with curvature. In fact, the product $\mu^2_{A^{\mathit{mod}}}$ does not actually change, since it is defined in a way which does not involve $\mu^*_A$. Concretely, this means that we have a map $$\label{eq:gamma-for-modules}
\Gamma^{\mathit{mod}}: \mathit{CC}(A,A) \longrightarrow \mathit{CC}(A^{\mathit{mod}},A^{\mathit{mod}})$$ such that $g^{\mathit{mod}} = \Gamma^{\mathit{mod}}(g)$ has only two nontrivial components $$\label{eq:gamma-mod}
\begin{aligned}
& g^{\mathit{mod},0} \in \mathit{hom}_{A^{\mathit{mod}}}^{|g|}(M_0,M_0), \\
& g^{\mathit{mod},1}: \mathit{hom}_{A^{\mathit{mod}}}(M_0,M_1) \longrightarrow \mathit{hom}_{A^{\mathit{mod}}}(M_0,M_1)[|g|-1].
\end{aligned}$$ These are given by $$\label{eq:gamma-formula}
\begin{aligned}
(g^{\mathit{mod},0})^d(m,a_{d-1},\dots,a_1) & = \sum_{ij} (-1)^\ast \, \mu_{M_0}^{d-j+1}(m,\dots, g^j(a_{i+j},\dots,a_{i+1}),\dots,a_1), \\
(g^{\mathit{mod},1})(b)^d(m,a_{d-1},\dots,a_1) & = \sum_{ij} (-1)^\ast \, b^{d-j+1}(m,\dots,g^j(a_{i+j},\dots,a_{i+1}),\dots,a_1), \\
& \ast = (|g|-1)(|a_{i+1}|+\cdots+|a_{d-1}|+|m|+d-i-1)+|g|.
\end{aligned}$$

There is a restriction map from the Hochschild complex of $A^{\mathit{mod}}$ to that of its full subcategory $A^{\mathit{perf}}$. One can accordingly restrict and get a map $\Gamma^{\mathit{perf}}$, which is of course defined by the same formulae. Another manifestation of derived invariance of Hochschild cohomology says that 