As in §2.5, in the embedded case, the Fukaya category ${\mathbin{\mathscr{F}}}(M)$ has objects $(L,E,b)$ for $(L,E)$ an embedded Lagrangian brane and $b$ a bounding cochain, but the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ has objects twisted complexes, consisting of objects $(L_{1},E_{1},b_{1}),\ldots,(L_{n},E_{n},b_{n})$ in ${\mathbin{\mathscr{F}}}(M)$ together with $b_{ij}\in CF^{*}\bigl((L_{i},E_{i}),(L_{j},E_{j})\bigr)$ for $1\leqslant\penalty i<j\leqslant\penalty n$ satisfying an equation.

In the immersed case, we can regard such a twisted complex as a single object $(L,E,b)$ in ${\mathbin{\mathscr{F}}}(M)$, where $L$ is the disjoint union $L_{1}\amalg\cdots\amalg L_{n}$, considered as a single immersed Lagrangian, $E|_{L_{i}}=E_{i}$, and $b$ is a bounding cochain for $(L,E)$ built from $b_{1},\ldots,b_{n}$ and $b_{ij}$ for $i<j$. Thus there is no need to add twisted complexes, and we can suppose all objects of $D^{b}{\mathbin{\mathscr{F}}}(M)$ are of the form $(L,E,b)$.

The idempotent completion $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ of $D^{b}{\mathbin{\mathscr{F}}}(M)$ as in §2.5 could still include objects which are direct summands of some $(L,E,b)$, but do not have a good geometric interpretation. However, in §3.1 we will conjecture that in the situation we are interested in, $D^{b}{\mathbin{\mathscr{F}}}(M)$ is already idempotent complete, so that we can take all objects of $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ to be of the form $(L,E,b)$.