In this talk we explain how we resolved the problem of extending normal coactions on non-self-adjoint algebras to their C*-envelope, and provide an application of our result to operator algebras arising from small categories in the sense of Spielberg. Our work yields generally applicable tools for computing the C*-envelope, recovers the main result in a recent breakthrough paper of Sehnem, and answers an open problem considered in the works of Kakariadis, Katsoulis, Laca and X. Li. We apply our main result to show that the C*-envelope of an operator algebras associated with groupoid-embeddable small category coincides with the boundary quotient C*-algebra of the small category, answering another problem pose by X. Li.