Let $\mathcal{C}$ be a $c$-unital $A\_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A\_\infty$-category, then there is a $c$-unital $A\_\infty$- functor 
$$Tw: fun(\mathcal{C},\mathcal{A})\rightarrow fun(Tw \mathcal{C}, Tw\mathcal{A})\rightarrow fun(Tw \mathcal{C}, \mathcal{A}),$$
See (3.26) in section (3n) of Seidel's book "Fukaya cateogories and Picard-Lefschetz Theory". Here $fun(\mathcal{C},\mathcal{A})$ is the $A\_\infty$-category of $c$-unital functors from $\mathcal{C}$ to $\mathcal{A}$, $Tw\mathcal{A}$ means the category of twisted complexes of $\mathcal{A}$ and the latter morphism uses the quasi-equivalence between $Tw\mathcal{A}$ and $\mathcal{A}$. 


There is also the restriction functor
$$\mathcal{I}^\*: fun(Tw \mathcal{C}, \mathcal{A})\rightarrow fun(\mathcal{C},\mathcal{A}).$$
My question is: are the two functors $Tw$ and $\mathcal{I}^\*$ in any sense inverse to each other?


My instinct is that if two functors $\mathcal{G}, \mathcal{H}\in fun(Tw\mathcal{C},
\mathcal{A})$ have $\mathcal{I}^\*\mathcal{G}\cong \mathcal{I}^\*\mathcal{H}$ in $H^0(fun(\mathcal{C},\mathcal{A}))$, then $\mathcal{G}\cong \mathcal{H}$ in $H^0(fun(Tw\mathcal{C},\mathcal{A}))$. But this seems to be too strong. 


For example, take
$\mathcal{A}=Ch$, the dg category of cochain complexes, I would expect 
$$\mathcal{I}^\*: mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$$ 
is cohomologically full and faithful. But I only see result like Lemma 3.36 in Seidel's
book which says
$$\tilde{l}: Tw(\mathcal{C})\rightarrow mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$$
is cohomologically full and faithful.


Thanks!



Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful?
This seems to me to be unproblematic (if messy to prove). Try if this does the job:
Proposition 12.15(2) in Bespalov-Lyubashenko-Manzyuk, Pretriangulated A\_infty-categories


