It is desirable to take Floer cohomology of $L$ with itself. One major feature of $HF^{*}(L,L)$ is that it contains units, at least at cohomological level.

One challenge to implement self Floer cohomology is that $L$ is not transverse to itself, so the Cauchy-Riemann equation needs perturbation. There are many frameworks to address this problem, and one idea dating back to Floer is to use the Hamiltonian invariance of Floer cohomology, to think of self Floer cohomology via $HF^{*}(L,L)\simeq HF^{*}(L,\phi_{\epsilon H}(L))$ where $\phi_{\epsilon H}$ is the time one flow of the small generic Hamiltonian $\epsilon H$ [9, section 1.6]. For $\epsilon\ll 1$, the Lagrangian $\phi_{\epsilon H}(L)$ can be identified as a graph over $L$ inside $T^{*}L$, the transverse intersection $L\cap\phi_{\epsilon H}(L)$ are the critical points of $H|_{L}$, and a suitable setup of the Floer trajectories (65) can be identified as Morse flowlines of $H$. Thus $HF^{*}(L,L)$ is isomorphic to the Morse cohomology of $L$, so $HF^{*}(L,L)\simeq H^{*}(L)$. In the exact case, the ring structure on $HF^{*}(L,L)$ defined from perturbed holomorphic triangles agrees with the cup product ring structure on $H^{*}(L)$. The unit can be represented by the Morse generator of $H^{0}(L)$, or more non-perturbatively via the Piunikhin-Salamon-Schwarz map.

It takes some effort to promote the self Floer cohomology to the Fukaya category framework, and ensure the consistency in the perturbation schemes (cf. Auroux [9, section 2.1] for a sketch and Seidel [69] for details). In applications it is often more convenient to avoid Hamiltonian perturbations as much as possible.