Siebert's formula for virtual pullbacks

For a DM-type morphism of algebraic stacks $f\text{\hspace{0.17em}⁣}:X\to Y$ with a relative perfect obstruction theory $\mathbb{E}\to {\mathbb{L}}_{X\mathrm{/}Y}$ having a global resolution, we give a formula for the induced virtual pullback in terms of $\mathbb{E}$ and $f$. We apply our formula to prove functoriality properties of the virtual fundamental class in Gromov-Witten theory.