A more Floer theoretic argument that $\mathcal{C}_{1}+\mathcal{C}_{2}$ is homologous to $\mathcal{C}_{3}$, which does not appeal to $H_{n+1}(X)=0$ directly, can be sketched as follows. We assume $L_{0},L_{0}^{\prime},L$ are three unobstructed Lagrangians mutually isomorphic in $D^{b}Fuk(X)$, and $HF^{-1}(L_{0},L_{0})=0$. Of course, the self Floer cohomologies of $L_{0},L_{0}^{\prime},L$ are all isomorphic, and $HF^{-1}=0$ is a necessary condition if the $D^{b}Fuk(X)$ class admits any almost calibrated representative at all. We consider $\alpha\in CF^{0}(L_{0},L_{0}^{\prime}),\beta\in CF^{0}(L_{0}^{\prime},L),\gamma\in CF^{0}(L,L_{0})$ representing the generators of $HF^{0}$, such that at the level of Floer cohomology

For simplicity we first assume almost calibratedness, so that $CF^{-1}=0$, and there is no ambiguity for these generators. Notice the compositions $\beta\circ\alpha,\gamma\circ\beta,\alpha\circ\gamma$ provide generators of $HF^{0}(L_{0},L)$, $HF^{0}(L_{0}^{\prime},L_{0})$, $HF^{0}(L,L_{0}^{\prime})$. Consider the $n$-dimensional moduli spaces $\tilde{\mathcal{M}}$ of holomorphic discs with corners at $\alpha,\beta,\gamma$ and the self intersection points corresponding to the bounding cochains. The corresponding universal family $\tilde{\mathcal{C}}$ provides an $(n+2)$-dimensional current, whose boundary comes from disc bubbling and disc breaking. Most of the boundary contributions are eliminated by the Mauer-Cartan equation of the bounding cochains, the closedness of $\alpha,\beta,\gamma$, and support dimension reasons, and only three boundary contributions survive. These are the $(n+1)$-dimensional bordism currents between $L_{0},L_{0}^{\prime}$ (resp. $L_{0}^{\prime},L$ and $L,L_{0}$) constructed from the universal family of holomorphic curves associated to the generators $\alpha,-\gamma\circ\beta$ (resp. $\beta,-\alpha\circ\gamma$ and $\gamma,\beta\circ\alpha$). We can identify these as $\mathcal{C}_{2},\mathcal{C}_{1},-\mathcal{C}_{3}$. The upshot is that Floer theory explicitly provides the $(n+2)$-dimensional current that exhibits the homological relation between $\mathcal{C}_{1}+\mathcal{C}_{2}$ and $\mathcal{C}_{3}$.

In general without assuming almost calibratedness, then $CF^{-1}$ can be nonzero. Then we need some extra $n$-dimensional moduli spaces to account for the non-uniqueness of cohomological representatives of $HF^{0}$, an issue quite similar to section 3.1.2. A subtle new issue is that the moduli space $\tilde{\mathcal{M}}$ receives new boundary contributions involving the $m_{3}^{b}$ products (this shorthand notation indicates the presence of bounding cochain elements, cf. (73)) of $\alpha,\beta,\gamma$. The three cyclic permutations of $\alpha,\beta,\gamma$ produce three $m_{3}^{b}$ products, which are elements in $CF^{-1}(L_{0},L_{0}),CF^{-1}(L_{0}^{\prime},L_{0}^{\prime})$ and $CF^{-1}(L,L)$ respectively, and the $(n-1)$-dimensional moduli of polygons with one corner at the $CF^{-1}$ intersections and the other corners at bounding cochain elements contribute to $\partial\tilde{\mathcal{C}}$. Now by the $A_{\infty}$ relation, and the closedness of $\alpha,\beta,\gamma$,

Writing $m_{2}^{b}(\gamma,m_{2}^{b}(\beta,\alpha))=1_{L_{0}}+m_{1}^{b}(\delta_{1})$ and $m_{2}^{b}(m_{2}^{b}(\gamma,\beta),\alpha)=1_{L_{0}}+m_{1}^{b}(\delta_{2})$, we see $m_{3}^{b}(\gamma,\beta,\alpha)+\delta_{1}-\delta_{2}$ is $m_{1}^{b}$-closed, so by the assumption that $HF^{-1}(L_{0},L_{0})=0$, it is in fact $-m_{1}^{b}(\epsilon_{1})$ for some $\epsilon_{1}\in CF^{-2}(L_{0},L_{0})$. We can then produce an $n$-dimensional moduli space, from polygons with a corner at $\epsilon_{1}$, and other corners at the bounding cochain elements. Completely analogously, one can produce two other $n$-dimensional moduli spaces from $\epsilon_{2}\in CF^{-2}(L_{0}^{\prime},L_{0}^{\prime})$ and $\epsilon_{3}\in CF^{-2}(L,L)$. Combining the $(n+2)$-dimensional universal families over the $n$-dimensional moduli spaces, results in an explicit bordism current between $\mathcal{C}_{1}+\mathcal{C}_{2}$ and $\mathcal{C}_{3}$.