In our convention an immersed Lagrangian can have several components. Of particular interest is the case where $L$ is the union of transverse immersed Lagrangians $L_{1},\ldots L_{N}$ with bounding cochains $b_{1},\ldots,b_{N}$ respectively, and we have morphisms $b_{ij}\in CF^{1}((L_{i},b_{i}),(L_{j},b_{j}))$ for $i>j$. The key assumption here is that the morphisms only go in one direction from $L_{i}$ to $L_{j}$, not vice versa. We assume that $b=\sum b_{i}+\sum_{i>j}b_{ij}$ is a bounding cochain for the immersed Lagrangian $L$, and in particular all intersection points in $b_{ij}$ satisfy the Novikov positivity condition $f_{L_{i}}\geq f_{L_{j}}$. We can write out the Mauer-Cartan equation

$$m_{0}+m_{1}(b)+m_{2}(b,b)+\ldots=0$$

in component form: for any $i>j$,

$$\sum_{l}\sum_{k\leq l}\sum_{i=i_{0}>\ldots>i_{k}=j}m_{l}(b_{i_{k}},\ldots b_{i_{k}},b_{i_{k-1}i_{k}},\ldots,b_{i_{1}},\ldots b_{i_{1}},b_{i_{0}i_{1}},b_{i_{0}},\ldots,b_{i_{0}})=0.$$

The key observation is that this is precisely how one would define twisted complexes built on $L_{1},\ldots L_{N}$, in the presence of the bounding cochains $b_{1},\ldots b_{N}$ and the data $b_{ij}$, when no further degree shifts are involved (cf. the exact setting in section 6.1). In this sense, we say that ‘immersed Lagrangians geometrises twisted complexes’. In other words, if the unobstructed immersed Lagrangians are admitted into the Fukaya category, then there is no need to formally add twisted complexes.