\mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace ,  \tilde{J}_i) \subset \mathcal{J}_t(M_i, \mathrm{int}\left\lbrace \omega_i = \tilde{\omega}_i \right\rbrace ,  \tilde{J}_i)$$ of regular almost complex structures such that the differential is finite and satisfies $\partial ^2 = 0$. Therefore, the quilted Floer homology $HF (\underline{L})$ is well-defined for generic almost complex structures and Hamiltonian perturbations, and is independent on these choices, except eventually on the reference almost complex structure.

For the manifolds we will work with, the choice of the reference almost complex structure will not affect the construction since it will be chosen in a contractible space, see [@MW Remark 4.13].

#### **Grading**

The hypothesis on the minimal Maslov number allows one to define a relative ${{\mathbb{Z}}_{8}}$-grading on the chain complex: if $\underline{x}$ and $\underline{y}$ are two generalized intersection points, and $u$, $v$ two quilted Floer trajectories (not necessarily pseudo-holomorphic) connecting them, $I(u) = I(v)$ modulo 8. We then denote $I(\underline{x},\underline{y}) \in {{\mathbb{Z}}_{8}}$ the common quantity. The differential is then of degree $1$. It follows that $I(\underline{x},\underline{y})$ defines a relative grading on $HF(\underline{L})$.

Recall then the following result, which ensures the invariance of quilted Floer homology under embedded geometric composition:

([@MW Theorem 6.7])\[compogeom\] Let $\underline{L}$ be a generalized Lagrangian correspondence as before. Assume moreover 