$\Gamma^{\mathit{perf}}$ is a quasi-isomorphism. Next, $A^{\mathit{perf}}$ contains a full subcategory quasi-isomorphic to $A^{\mathit{tw}}$, and by restriction and the argument from , one gets a further map from the Hochschild cohomology of $A^{\mathit{perf}}$ to that of $A^{\mathit{tw}}$. The situation can be summarized in the commutative diagram $$\xymatrix{
\mathit{HH}(A,A)
\ar@/_1pc/[drr]^-{\Gamma^{\mathit{tw}}}_{{\cong}} \ar@/_3pc/[ddrr]^-{\Gamma^{\mathit{perf}}}_{{\cong}} \ar@/_6pc/[dddrr]^-{\Gamma^{\mathit{mod}}} \ar[rr]^-{\mathrm{id}} && \mathit{HH}(A,A) \\
&& \mathit{HH}(A^{\mathit{tw}},A^{\mathit{tw}}) \ar[u]_{{\cong}} \\
&& \mathit{HH}(A^{\mathit{perf}},A^{\mathit{perf}}) \ar[u]_{{\cong}} \\
&& \mathit{HH}(A^{\mathit{mod}},A^{\mathit{mod}}) \ar[u]
}$$ where the vertical arrows are restriction type maps. Other than derived invariance (which we will not discuss further, but see [@keller06 Section 5.4]), the only nontrivial point in this diagram is that the map $\Gamma^{\mathit{mod}}$ is compatible with restriction. To understand that, we have to look at for the Yoneda embedding: $$\label{eq:yoneda-functoriality}
\mathit{CC}(A,A) \longrightarrow \mathit{CC}(A,A^{\mathit{mod}}) \longleftarrow \mathit{CC}(A^{\mathit{mod}},A^{\mathit{mod}}).$$ Take $g \in \mathit{CC}(A,A)$, and consider its image under the first map in . For simplicity let’s look only at the constant term of this, which consists of an endomorphism $b \in \mathit{hom}_{A^{\mathit{mod}}}(Y^{\mathit{yon}},Y^{\mathit{yon}})$ for each $Y$, given by $$b^d(a_d,\dots,a_1) = \mu^{d+1}_A(g^0,a_d,\dots,a_1). \label{eq:1-cc}$$ On the other hand, we can take $g^{\mathit{mod}} \in \mathit{CC}(A^{\mathit{mod}},A^{\mathit{mod}})$ and pull it back to $\mathit{CC}(A,A^{\mathit{mod}})$ as in the second map in , which leads to another cochain $\mathit{CC}(A,A^{\mathit{mod}})$ with constant term $$\label{eq:2-cc}
\begin{aligned}
\tilde{b}^d(a_d,\dots,a_1) 