Using: https://arxiv.org/pdf/math/9910179.pdf as a reference...

My question involves spelling out explicitly the comment in 4.2 -

"Equivalently, the datum of an $A_\infty$-structure on a graded space $M$ is the datum of a differential $m_1$ on $M$ and of a morphism of $A_\infty$-algebras from $A$ to the opposite of the dg-algebra $Hom_k(M,M)$ (cf. 3.3)."

$Q1$: What is the $A_\infty$ structure on $Hom_k(M,M)^\text{op}$? (Is it just $m_1$ is the usual differential and $m_2$ is composition in opposite order with all higher $m_i$ vanishing for $i>2$..?) Could they possibly mean any other $A_\infty$-structure?

$Q2$: If my answer for Q1 is correct how does this imply that an $A_\infty$-module structure on $M$ is the same as an $A_\infty$ map $\alpha:A \to Hom_k(M,M)^\text{op}$ given that:

a map of $A_\infty$-algebras is the same as having

$$ \alpha_m:A^{\otimes m}\to Hom_k(M,M)^\text{op} $$

for all $m>0$ s.t.

$$ \sum(-1)^{r+st}\alpha_u(1^{\otimes r}\otimes m^A_s \otimes 1^{\otimes t})=\sum(-1)^{s}m^M_r(\alpha_{i_1} \otimes \cdots \otimes \alpha_{i_r} ) $$

where the left hand side is a sum over all decompositions $m=r+s+t$ where $u=r+1+t$, $s\geq1$, and $r,t\geq 0$. The right hand side is a sum over all $1\leq r \leq m$ and over all $i_1+\cdots +i_r=m$ and the sign on the right is given in the paper in 3.4.

$m^M_r$ are the higher multiplications on $Hom_k(M,M)$ that are referenced in Q1.

Ps. Obviously, there will have to be some use of tensor-hom adjunction, I just can't quite spell it out.