Let $X$ be a $n$ (complex) dimensional variety and let $\mathcal{F}$ be a coherent sheaf on $X$. Using the sheaf hom which I denote $\mathcal{H}om$, we can define the dual coherent sheaf as
$$\mathcal{F}^{\vee} = \mathcal{H}om(\mathcal{F}, \mathcal{O}_{X}).$$
With this definition, I'm hoping someone can explain how to compute the Chern character $\text{ch}(\mathcal{F}^{\vee})$ of the dual sheaf.
Given a class $v = \oplus_{i} v_{i} \in H^{*}(X, \mathbb{Q})$, I believe it is standard to define the dual class to be $v^{\vee} = \oplus_{i}(-1)^{i} v_{i} \in H^{*}(X, \mathbb{Q})$. This is, for example, done in Huybrechts and Lehn. The motivation behind this definition is supposed to be such that
$$\text{ch}^{\vee}(\mathcal{F}) = \text{ch}(\mathcal{F}^{\vee}).$$
Assuming someone can assist me in computing the Chern character of the dual sheaf, will it be consistent with this definition?