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Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{B}$) and suppose there is a natural transformation $T:F \longrightarrow G$. Is it always possible to lift $T$ to a natural transformation $\mathcal{T}:\mathcal{F}\longrightarrow \mathcal{G}$ ?

Another version of the same question I care about is if for every $X \in ob(\mathcal{A})$ I have a morphism $t^0_X \in hom_\mathcal{B}(\mathcal{F}(X),\mathcal{G}(X))$ such that in homology the classes represented by $t_X^0$ defines a natural transformation $[t]:F\longrightarrow G$, when is there a natural transformation $\mathcal{T}:\mathcal{F}\longrightarrow \mathcal{G}$ such that $[\mathcal{T}]=[t]$?

What I need is an explicit construction, not just existence.

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    $\begingroup$ Interesting question and welcome to MO Vincent — out of curiosity, what would an explicit construction give you that existence wouldn’t (aside from the obvious advantage of a better understanding of the machinery in play, but in the context of proving something further is there an advantage)? $\endgroup$
    – Alec Rhea
    Commented Mar 8, 2019 at 18:06
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    $\begingroup$ Hi Alec! I mean, existence would already be good news but in the end I'm interested in knowing about some structure being preserved by functors between different instances of the Fukaya category and I dont think I would be able to SEE that from just knowing the natural transformation exists $\endgroup$
    – Vincent L.
    Commented Mar 8, 2019 at 18:38
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    $\begingroup$ The answer is negative, and you can find an obstruction theory for such problems discussed in Seidel's book. For an example, try $A = C^*(S^2)$ and $B= C^*(S^3)$, and $F$ and $G$ respectively given by the Hopf map and the trivial map. $\endgroup$
    – Mohammed Abouzaid
    Commented Mar 8, 2019 at 19:35

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