For Fukaya categories there are functors naturally induced by symplectomorphisms. Twisted versions of symplectic homology (fixed point Floer homology), open-closed maps and bimodules can be defined. However I haven't seen a definition of twisted bimodule (i.e. graph bimodules) yet.
Given an $A_\infty$ category $(\mathcal{A},\mu_i)$, the diagonal bimodule is defined as $\Delta(A,B):=\mathcal{A}(A,B)$ with structure maps given by $\mu=\bigoplus \mu_{r|1|s}=\bigoplus \mu_{r+s+1}$.
Given an $A_\infty$ functor $\phi=\bigoplus \phi_i:\mathcal{A}\rightarrow \mathcal{A}$, it is natural to define the graph bimodule $\mathcal{A}_\phi(A, B):=\mathcal{A}(A,\phi B).$ However the structure maps apparently should involve $\phi_i$ and seems to be complicated. Is there a nice formula for it?
