\geq 3$ and one of the $a_i$ is a unit. Such modules form an $A_\infty$-category $A^{\mathit{mod}}$ (this is in fact a differential graded category, but we prefer to view it as an $A_\infty$-category with trivial higher order products, which entails slightly different sign conventions). A morphism $b \in \mathit{hom}_{A^{\mathit{mod}}}(M_0,M_1)$ consists of $$\label{eq:module-map}
\begin{aligned}
& b^1: M_0(X_0) \longrightarrow M_1(X_0)[|b|], \\
& b^2: M_0(X_1) \otimes \mathit{hom}_A(X_0,X_1) \longrightarrow M_1(X_0)[|b|-1], \\
& \dots
\end{aligned}$$ Again, we require strict unitality, which means that $b^d(m,a_{d-1},\dots,a_1) = 0$ whenever one of the $a_i$ is a unit.

The Yoneda embedding ([@fukaya01b] or [@seidel04 Section 1l]) is a canonical $A_\infty$-functor $A \rightarrow A^{\mathit{mod}}$. On objects, it maps $Y$ to the module $Y^{\mathit{yon}}$ with $Y^{\mathit{yon}}(X) = \mathit{hom}_A(X,Y)$ and $\mu^d_{Y^{\mathit{yon}}} = \mu^d_A$. The first level map on morphisms is $$\label{eq:yoneda1}
\begin{aligned}
& \mathit{hom}_A(Y_0,Y_1) \longrightarrow \mathit{hom}_{A^{\mathit{mod}}}(Y_0^{\mathit{yon}},Y_1^{\mathit{yon}}), \\
& a \longmapsto a^{\mathit{yon}}, \quad a^{\mathit{yon},d}(a_d,\dots,a_1) = \mu^{d+1}_A(a,a_d,\dots,a_1).
\end{aligned}$$

\[th:yoneda\] The map is a quasi-isomorphism.

Consider the map in inverse direction, taking a module homomorphism $b$ to the element $a = b^1(e_{Y_0})$. Composing the two in one way yields the identity map on $\mathit{hom}_A(Y_0,Y_1)$. The other composition is chain homotopic to the identity: an explicit homotopy is $$\label{eq:h-homotopy}
\begin{aligned}
& h: \mathit{hom}_{A^{\mathit{mod}}}(Y_0^{\mathit{yon}},Y_1^{\mathit{yon}}) \longrightarrow \mathit{hom}_{A^{\mathit{mod}}}(Y_0^{\mathit{yon}},Y_1^{\mathit{yon}})[-1], \\
