Fibers of toric morpisms

Question

Let $f: X(\Delta_1) \to X(\Delta_2)$ be a toric morphism of toric varieties, and let $\sigma \subset \Delta_2$ be a cone, then for any point in the corresponding orbit $x \in O(\sigma)$ the fiber $f^{-1}(x)$ is the same. Is there a way to describe this fiber in terms of fans? Is it true that the fibers are always toric varieties? In this case is there a way to see corresponding fan?