The Dehn-Nielsen-Baer Theorem for Bounded Surfaces

Let $\mathrm{\Sigma }$ be a bounded surface. We prove the Dehn-Nielsen-Baer theorem for bounded surfaces to show that the mapping class group of $\mathrm{\Sigma }$ is isomorphic to the automorphisms of the fundamental groupoid of $\mathrm{\Sigma }$ that fix loops around the boundary.