a_1 \otimes p) = \\
& \qquad \sum_i (-1)^{|p|+|a_1|+\cdots+|a_i|-i} \mu^{r-i+1}_M(m,a_r,\dots,a_{i+1}) \otimes a_i \cdots \otimes a_1 \otimes p \\ &
\qquad + \sum_{i,j} (-1)^{|p|+|a_1|+\cdots+|a_i|-i} m \otimes \cdots \otimes \mu_A^j(a_{i+j},\dots,a_{i+1}) \otimes a_i \otimes \cdots \otimes a_1 \otimes p \\ &
\qquad + \sum_i m \otimes \cdots \otimes a_{i+1} \otimes \mu_P^{i|1|0}(a_i,\dots,a_1,p).
\end{aligned}$$ There are similar formulae for $\mu^d_{\tilde{M}}$, $d > 1$, which are in fact simpler since they involve only $\mu_P^{*|1|d-1}$. Naturality of the tensor product with a fixed $P$ is expressed by a map $\mathit{hom}_{A^{\mathit{mod}}}(M_0,M_1) \rightarrow \mathit{hom}_{\tilde{A}^{\mathit{mod}}}(\tilde{M}_0,\tilde{M}_1)$ for $\tilde{M}_k = M_k \otimes_A P$. This takes $b$ to $b \otimes_A e_P$, given by $$\begin{aligned}
& (b \otimes_A e_P)^1(m_0 \otimes a_r \cdots \otimes a_1 \otimes p) = \\
& \qquad \qquad \qquad \sum_i (-1)^{|p|+|a_1|+\cdots+|a_i|-i} b^{r-i+1}(m_0,a_r,\dots,a_{i+1}) \otimes a_i \otimes \cdots \otimes p,
\end{aligned}$$ with vanishing higher order terms. The resulting convolution $A_\infty$-functor (in fact a dg functor, since it has no higher order terms) between module categories will be denoted by $K_P$. So far, we have skirted the issue of whether $M \otimes_A P$ is really an object of $A^{\mathit{mod}}$ as defined, meaning whether it satisfies the cohomological finiteness condition. This fails in general, but it will hold if $M$ is perfect (based on the 