In this talk we will investigate an ordinary rank-two case of local-global compatibility. Roughly speaking, the local-global compatibility conjecture predicts an isomorphism between the Weil--Deligne representations obtained from certain automorphic representations in two ways. We will explain how these Weil--Deligne representations are obtained, and then present some known cases of local-global compatibility (which inspired this work). We will introduce the idea of automorphy lifting, and sketch the key steps in the proof of the ordinary case.