= & -\sum_{i+j<d} (-1)^\ast \mu^{d-j+1}_A(a_d,\dots,g^j(a_{i+j},\dots,a_{i+1}),\dots,a_1), \\ & \ast = (|g|-1)(|a_{i+1}|+\cdots+|a_d|+d-i).
\end{aligned}$$ Assuming that $g$ is a Hochschild cocycle, we can write $$\begin{aligned} &
\tilde{b}^d(a,a_{d-1},\dots,a_1) - b^d(a,a_{d-1},\dots,a_1) = \\ &
\qquad \sum_{i<d} (-1)^{(|g|-1)(|a_{i+1}| + \cdots + |a_d| + d - i)} \mu^{i+1}_A(g^{d-i}(a_d,\dots,a_{i+1}),\dots,a_1) \\  & \qquad + \sum_{i,j} (-1)^{|g| + |a_1| + \cdots + |a_i| - i + (|g|-1)(|a_1|+\cdots+|a_d|-d)} g^{d-j+1}(a_d,\dots,\mu_A^j(a_{i+j},\dots,a_{i+1}),\dots,a_1).
\end{aligned}$$ Quite obviously, this is the coboundary of another endomorphism of $Y^{\mathit{yon}}$ of degree $|g|-1$, given by $(a_d,\dots,a_1) \mapsto (-1)^{|g|(|a_1|+ \cdots + |a_d|-d)+1}g^d(a_d,\dots,a_1)$, hence zero on cohomology. The general computation is similar.

Bimodules\[subsec:bimodules\]
-----------------------------

$A_\infty$-bimodules have already made a brief appearance before, but we will now consider them in a little more detail. Let $A$ and $\tilde{A}$ be $A_\infty$-categories over $R$, with the usual unitality and properness assumptions. An [*$(A,\tilde{A})$-bimodule with finite cohomology*]{} $P$ assigns to any pair of objects $(X,\tilde{X}) \in \mathit{Ob}\, A \times \mathit{Ob}\,\tilde{A}$ a graded vector space $P(\tilde{X},X)$, which comes with structure maps $$\label{eq:bimodule-structure}
\begin{aligned}
& \mu_P^{s|1|t}: \mathit{hom}_A(X_{s-1},X_s) \otimes \cdots \otimes \mathit{hom}_A(X_0,X_1) \otimes P(\tilde{X}_t,X_0) \\ & \qquad \qquad \qquad \qquad \otimes \mathit{hom}_{\tilde{A}}(\tilde{X}_{t-1},\tilde{X}_t) \otimes \cdots \otimes \mathit{hom}_{\tilde{A}}(\tilde{X}_0,\tilde{X}_1) \longrightarrow P(\tilde{X}_0,X_s)[1-s-t]
\end{aligned}$$ for all $s,t \geq 0$, satisfying analogues of the $A_\infty$-module equation. We assume that the cohomology groups $H(P(\tilde{X},X),\mu_P^{0|1|0})$ are of 