We propose a formalization of HO$\pi$ in Coq, a process calculus where messages carry processes. Such a higher-order calculus features two very different kinds of binder: process input, similar to $\lambda$-abstraction, and name restriction, whose scope can be expanded by communication. We formalize strong context bisimilarity and prove it is compatible, i.e., closed under every context, using Howe’s method, based on several proof schemes we developed in a previous paper.