Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.

Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack. Let $\mathbf{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a scheme. Let $\mathbb{F}_2$ be the Hirzebruch surface of degree 2.

$\mathbf{P}(1,1,2)$ has a unique singularity and $\mathbb{F}_2$ is the minimal resolution.

From derived McKay correspondence, we have an equivalence of derived categories $$ \Phi:D^b(\text{Coh}(\mathbb{P}(1,1,2))) \overset{\simeq}{\rightarrow} D^b(\text{Coh}(\mathbb{F}_2)). $$

**Question**

How is this functor $\Phi$ defined?

For example, what are the images of canonical line bundles $\mathscr{O}_{\mathbb{P}(1,1,2)}(i)$ by $\Phi$ ? Of course, I am also interested in the image of any other sheaf.

Any comment is welcome. Thank you.