For simplicity we will assume that all local systems are trivial and have rank one. The general case is completely similar.

We are going to construct the following chain of $A_{\infty}$-equivalences connecting $DR(Y)$ and $M(Y)$:

$$DR(Y)\hookrightarrow DR_{0}(Y)\hookleftarrow DR_{0}^{tr}(Y)\hookleftarrow DR_{0}^{tr,\Pi}(Y))\leftarrow M(Y)$$

Classes of objects of all these categories will be the same, and all functors will be identical on objects.

The $A_{\infty}$-pre-category $DR_{0}(Y)$ is in fact a dg-category, i.e. all sequences of objects are transversal, compositions $m_{k}$ vanish for $k\geq 3$ and it has strict identity morphisms. The space $Hom_{DR_{0}(Y)}(f_{0},f_{1})$ is defined as $\varinjlim_{\delta\to 0}\Omega^{\ast}(Y\times(0,\delta))=\Omega_{0}^{\ast}(Y).$ Clearly the space of morphisms does not depend on objects. Using the wedge product of differential forms we make $DR_{0}(Y)$ into a dg-category over the field ${\bf C}$. There is a natural functor $DR(Y)\to DR_{0}(Y)$, which is the identity map on objects. On morphisms it is the natural embedding of $\Omega^{\ast}(Y)$ as the subspace of forms on $Y\times(0,\delta)$, which are pullbacks of forms on $Y$. Clearly it establishes an equivalence of $A_{\infty}$-categories.

The $A_{\infty}$-pre-category $DR_{0}^{tr}(Y)$ is defined as the full subcategory of $DR_{0}(Y)$, and it differs from the latter only by the choice of transversal sequences. Namely, we use the same notion of transversality in $DR_{0}^{tr}(Y)$ as in the Morse category.

The next $A_{\infty}$-pre-category $DR_{0}^{tr,\Pi}(Y)$ is obtained from $DR_{0}^{tr}(Y)$ by applying homological perturbation theory. For any two transversal objects $f_{0},f_{1}$ of $DR_{0}^{tr}(Y)$ we define $Hom_{DR_{0}^{tr,\Pi}(Y)}(f_{0},f_{1})$ as $\Pi_{f_{0},f_{1}}(\Omega^{\ast}_{0}(Y))$. Here $\Pi_{f_{0},f_{1}}$ is the projector $\Pi$ corresponding to the Morse function $f_{0}-f_{1}$, it was described at the end of the previous subsection. We also have homotopies $H_{f_{0},f_{1}}$ associated with $f_{0}-f_{1}$. Then formulas of homological perturbation theory (summation over trees) give rise to an $A_{\infty}$-pre-category $DR_{0}^{tr,\Pi}(Y)$ and an equivalence $DR_{0}^{tr,\Pi}(Y)\to DR_{0}^{tr}(Y)$.

The last functor $\Psi:M(Y)\to DR_{0}^{tr,\Pi}(Y))$ will have no non-trivial higher components $\Psi_{n}$ for $n\geq 2$. The first component $\Psi_{1}$ of it is a linear map

$$\Psi_{1}:Hom_{M(Y)}(f_{0},f_{1})\to Hom_{DR_{0}^{tr,\Pi}(Y)}(f_{0},f_{1})$$

for every transversal pair $(f_{0},f_{1})$. Recall that $Hom_{M(Y)}(f_{0},f_{1})$ has a basis $\{[x]\}$ labeled by critical points $x\in Cr(f_{0}-f_{1})$. We define $\Psi_{1}([x])$ as $R([S_{x}])$. It is clear that $\Psi_{1}$ gives a quasi-isomorphism of complexes for every transversal pair $(f_{0},f_{1})$.

Now, we claim that $\Psi$ is an $A_{\infty}$-functor. This means that $\Psi_{1}$ maps all higher compositions in $M(Y)$ to higher compositions in $DR_{0}^{tr,\Pi}(Y)$. This follows directly from the descriptions of higher compositions in both categories in terms of planar trees and the lemma in the previous subsection. Indeed, the number of functions in any given sequence is finite. For all sufficiently small $\delta$ every summand in the formula for $m_{k}^{M(Y)}$, corresponding to a binary tree $T$, coincides with the summand for $m_{k}^{M(Y)}$ corresponding to the same $T$ (we can assume that $\delta$ is so small that the part 2) of the Lemma can be applied). The theorem is proved. $\blacksquare$