\cap M_0\times L_{12}$ in $M_0\times  M_2$.

When this criterion is fulfilled, not only $L_{01} \circ L_{12}$ is a Lagrangian correspondence, but quilted Floer homology also behaves well in this situation: we will see in Theorem \[compogeom\] that, assuming $L_{01} \circ L_{12}$ is embedded, and some extra hypothesis, $$HF( \cdots,  L_i, L_{i+1}, \cdots ) \simeq HF( \cdots,  L_i \circ L_{i+1}, \cdots ).$$

Wehrheim and Woodward then define the extended symplectic category $ \mathrm{Symp}_\tau^ \#$ (see [@WWfft Def. 3.1.7]) as the category whose objects are symplectic manifolds (with some extra monotonicity conditions), and morphisms equivalence classes of generalized Lagrangian correspondences (also satisfying extra conditions), where the equivalence relation is generated by: $$( \cdots,  L_i, L_{i+1}, \cdots ) \sim ( \cdots,  L_i \circ L_{i+1}, \cdots ),$$ provided $L_i \circ L_{i+1}$ is embedded, and some extra conditions ensuring Floer homology is well-defined.

It will follow from Cerf theory that the generalized Lagrangian correspondences we will construct are equivalent to length 2 generalized Lagrangian correspondences. This facts holds for every generalized Lagrangian correspondence, as observed by Weinstein in [@Weinsteinremark].

In order to fit in the framework of [@MW Assumption 2.5] required for defining symplectic instanton homology, we will adapt the definition of $ \mathrm{Symp}_\tau^ \#$ 