Let $X$ be a smooth algebraic variety. In addition, assume it to be a compact Calabi-Yau, $\omega_{X} \cong \mathcal{O}_{X}$. I have a coherent sheaf $\mathscr{F}$ on $X$ which is torsion-free, but not locally-free and it is resolved by vector bundles $\mathscr{A}$ and $\mathscr{B}$ as
$$0 \to \mathscr{A} \to \mathscr{B} \to \mathscr{F} \to 0.$$
Now, an object in the derived category $D^{b}(\text{Coh}(X))$ in which I am particularly interested, is not the complex $[\mathscr{A} \to \mathscr{B}]$, but rather $[\mathscr{B}^{\vee} \to \mathscr{A}^{\vee}]$. Here, the duals are well defined since these are vector bundles, so we simply take sheaf Hom into $\mathcal{O}_{X}$.
My question is, quite simply, is the complex $[\mathscr{B}^{\vee} \to \mathscr{A}^{\vee}]$ related (in the derived category) in any way to $\mathscr{F}^{\vee}$, under some suitable definition of duality for torsion-free sheaves? Should I be looking into Verdier duality here?
Although I know very little about it, I've seen that Verdier duality of a complex in the derived category requires applying Hom into the dualizing complex $\omega_{X}$. Assuming $X$ is smooth, reduced I'm guessing this dualizing complex is simply zero except with the canonical bundle in degree zero. Since I am assuming my $X$ is Calabi-Yau, I am wondering if this gives anything special. Like, in this case is the complex $[\mathscr{A} \to \mathscr{B}]$ Verdier dual to $[\mathscr{B}^{\vee} \to \mathscr{A}^{\vee}]$?
The complex $B^\vee \to A^\vee$ is the derived dual of $F$. Its relation to the standard dual is $H^0(B^\vee \to A^\vee) \cong F^\vee$, $H^1(B^\vee \to A^\vee) \cong Ext^1(F,O_X)$ (here $H^i$ stand for the cohomology sheaves of a complex, and $Ext^1$ is a local $Ext$).
In plain words, the same can be phrased by saying that there is an exact sequence $$ 0 \to F^\vee \to B^\vee \to A^\vee \to Ext^1(F,O_X) \to 0. $$