Hyperelliptic linear system on a K3 surface ($\mathrm{deg}{\phi }_{|D|}=2$)

For a basepoint-free nef big divisor $D$, this means that the morphism given by its complete system is generically two-to-one onto its image. Equivalently, its general smooth curve is hyperelliptic. A double cover of ${\mathbb{P}}^2$ branched over a smooth sextic, with the pullback of $\mathcal{O}(1)$, gives the degree-two example.