$j_k$. The boundary components of $S_k$ will be indexed by a set $\mathcal{B}(S_k)$: $\partial S_k = \bigcup_{b\in\mathcal{B}(S_k) }{I_{k,b}}$.

2.  a collection $\mathcal{S}$ of *seams*, consisting of pairwise disjoints two-element subsets: $ \sigma \subset \bigcup_{k=1}^{m}\bigcup_{b\in\mathcal{B}(S_k)} I_{k,b}$, and for each $\sigma = \lbrace  I_{k,b}, I_{k',b'}\rbrace$, a real analytic diffeomorphism $\varphi_\sigma \colon I_{k,b} \rightarrow I_{k',b'}$.

Let $\underline{S}$ be a quilted surface as before, and $\underline{M} = (M_k)_{k=1\cdots m}$ a collection of symplectic manifolds, one for each patch, and $$\underline{L} = \left(  L_\sigma \subset M_k ^- \times M_{k'} , L_{k,b} \subset M_k\right)$$ a collection of Lagrangian correspondences, one associated to each seam $\sigma = \lbrace  I_{k,b}, I_{k',b'}\rbrace$, and Lagrangian submanifolds, one for each $I_{k,b}$ which is not contained in any seam. A *(pseudo-holomorphic) quilt* $  \underline{u}\colon   \underline{S} \rightarrow \left(\underline{M}, \underline{L} \right) $ is a collection of (pseudo-holomorphic, provided the $M_k$ are endowed with almost-complex structures) maps $u_i\colon S_i\rightarrow M_i$ satisfying the following seam and boundary conditions: $$(u_k(x), u_k'(\varphi_\sigma (x)) ) \in L_\sigma,  x \in I_{k,b}  ,$$ $$u_k(x) \in L_{k,b}, x \in I_{k,b} .$$

These data can be summarized in a diagram as in figure \[cylinder\]:

1.  We will refer to $I_{k,b}$ as a boundary of $\underline{S}$ if it is not contained in any seam.

