Consider a smooth projective variety of ample $\omega\_X$, how can I quickly see that
$$\textbf{Coh}(X)=\{\mathcal{F}^{\bullet}\mid\text{Hom}(\omega\_X^{\otimes i},\mathcal{F}^\bullet[n])=0\text{ for } n \neq0\text{ and }i\ll0\}\subset D^b(X)$$
This is used in the proof of [Bondal—Orlov reconstruction theorem](https://homepage.mi-ras.ru/%7Eorlov/papers/Compositio2001.pdf).



A characterization on coherent sheaves inside $D^b(X)$
First, one has
$$
\mathrm{Hom}(\omega\_X^{\otimes i}, F[n]) \cong \mathbb{H}^n(X, F \otimes \omega\_X^{\otimes -i}).
$$
Next, there is the hypercohomology spectral sequence
$$
H^q(X, \mathcal{H}^p(F) \otimes \omega\_X^{\otimes -i}) 
\Rightarrow 
\mathbb{H}^{p+q}(X, F \otimes \omega\_X^{\otimes -i}).
$$
Now, since $\omega\_X$ is ample and $i \ll 0$, for each (of the finite number) of nonzero sheaves $\mathcal{H}^p(F)$ the twist $\mathcal{H}^p(F) \otimes \omega\_X^{\otimes -i}$ has no higher cohomology and is globally generated, therefore the spectral sequence degenerates and gives
$$
\mathbb{H}^{n}(X, F \otimes \omega\_X^{\otimes -i}) \cong
H^0(X, \mathcal{H}^n(F) \otimes \omega\_X^{\otimes -i}).
$$
Thus, the left hand side vanishes for all $n \ne 0$ if and only if $\mathcal{H}^n(F) = 0$ for all such $n$.


