Graph morphisms as groupoid actors

We describe proper actors from the underlying groupoid of a graph C*-algebra to another \'etale groupoid in terms of bisections. This allows to understand graph morphisms and the *-homomorphisms that they induce more conceptually. More generally, we describe actors from the groupoid model of a groupoid correspondence to any \'etale groupoid. This also covers the groupoids associated to self-similar groups and self-similar graphs, among others.