The whole approach here is parallel to the one from Section 6, so we will omit the details. In the previous subsection we defined the semigroup $\psi^{t}$ acting on the sections with compact support $\Gamma_{c}(P(L_{1},L_{2}),\underline{Hom}^{alg}(E_{1},E_{2}))$. This action corresponds to the action of the semigroup $\phi^{t}$ on the space of morphisms $Hom_{{\cal C}(Y)}(E_{1},E_{2})$. Similarly to the case of Morse theory (Section 6) one proves the following result.

The limit is not difficult to describe in terms of the gradient flow generating $\psi^{t}$. Using the fact that $\psi^{t}$ moves the spectrum of a morphism to $-\infty$, one can prove similarly to the Section 6 that the limit $\psi^{\infty}(\beta)$ belongs to a finite-dimensional ${{\bf C}}_{\varepsilon}$-vector space generated by the distributions corresponding the unstable manifolds $U_{x}\subset P(L_{1},L_{2}),x\in L_{1}\cap L_{2}$. Clearly, the map $\beta\mapsto\psi^{\infty}(\beta)$ extends to the completion with respect to the filtration. It descends to the map $\alpha\mapsto\phi^{\infty}(\alpha)$, where $\alpha\in Hom_{{\cal C}(Y)}(E_{1},E_{2})$. The image of $\phi^{\infty}$ belongs to the space isomorphic to $Hom_{FO(X^{\vee})}((L_{1},\rho_{1}),(L_{2},\rho_{2}))$.

We can repeat the arguments from the Morse theory (see Section 6). We define the $A_{\infty}$-pre-category ${\cal C}_{unram,0}(Y)$ similarly to the category $DR_{0}(Y)$ from Section 6. It is $A_{\infty}$-equivalent to ${\cal C}_{unram}(Y)$. By definition the spaces of morphisms of ${\cal C}_{unram,0}(Y)$ are dg-modules over the dg-algebra ${{\bf C}}_{\varepsilon}\widehat{\otimes}\Omega_{0}^{\ast}$, where $\Omega_{0}^{\ast}$ is the dg-algebra of germs of differential forms at $0\in{{\bf R}}_{\geq 0}$. Compositions of morphisms in ${\cal C}_{0}(Y)$ are linear with respect to the dg-module structure. Imposing transversality conditions on ${\cal C}_{unram,0}(Y)$ to be the same as in $FO(X^{\vee})$, we obtain an $A_{\infty}$-equivalent $A_{\infty}$-pre-category ${\cal C}_{unram,0}^{tr}(Y)$.

Using homological perturbation theory (projectors and homotopies are defined by means of the semigroup) similarly to Section 6, we construct an analog of the category $DR_{0}^{tr,\Pi}(Y)$. It is an $A_{\infty}$-pre-category denoted by ${\cal C}_{unrum,0}^{tr,\Pi}(Y)$, with the spaces of morphisms which are completed tensor products of $\Omega_{0}^{\ast}$ with finite-dimensional ${{\bf C}}_{\varepsilon}$-vector spaces, spanned by the “smoothenings” of the unstable currents $[U_{x}]$ (cf. Section 6). By definition, it has the same transversality conditions as the category $FO(X^{\vee})$, and the spaces of morphisms are naturally quasi-isomorphic to the corresponding spaces of morphisms in $FO(X^{\vee})$ (compare with the Section 6.6). Similarly to the Section 6 we see that the $A_{\infty}$-structure on ${\cal C}_{unram,0}^{tr,\Pi}(Y)$ is equivalent to the one on $FO(X^{\vee})$. More precisely, we have a natural map from the space $Hom_{FO(X^{\vee})}((L_{1},\rho_{1}),(L_{2},\rho_{2}))$ (it is defined in terms of the Morse theory) to the space $Hom_{{\cal C}_{unram,0}^{tr,\Pi}(Y)}(F(L_{1},\rho_{1}),F(L_{2},\rho_{2}))$ (it is defined in terms of de Rham differential forms on $Y$). Thus we have defined the mirror symmetry functor $F$ on morphisms. Let us call the corresponding map $\nu_{X_{1},X_{2}}$ for $X_{i}=(L_{i},\rho_{i}),i=1,2$. The proof of the following proposition is similar to its analog from Section 6.6.

Thus, $A_{\infty}$-pre-categories ${\cal C}_{unram,0}^{tr,\Pi}(Y)$ and $FO(X^{\vee})$ are equivalent. By the same arguments as in the Morse theory section we see that ${\cal C}_{unram}(Y)$ and ${\cal C}_{unram,0}^{tr,\Pi}(Y)$ are also equivalent. Finally, applying functor $F$, we get our main result.

This is the version of homological mirror symmetry we promised to prove.