  Compositional methods are central to the development and verification of
software systems. They allow to break down large systems into smaller
components, while enabling reasoning about the behaviour of the composed
system. For concurrent and communicating systems, compositional techniques
based on behavioural type systems have received much attention. By abstracting
communication protocols as types, these type systems can statically check that
programs interact with channels according to a certain protocol, whether the
intended messages are exchanged in a certain order. In this paper, we put on
our coalgebraic spectacles to investigate session types, a widely studied class
of behavioural type systems. We provide a syntax-free description of
session-based concurrency as states of coalgebras. As a result, we rediscover
type equivalence, duality, and subtyping relations in terms of canonical
coinductive presentations. In turn, this coinductive presentation makes it
possible to elegantly derive a decidable type system with subtyping for
$\pi$-calculus processes, in which the states of a coalgebra will serve as
channel protocols. Going full circle, we exhibit a coalgebra structure on an
existing session type system, and show that the relations and type system
resulting from our coalgebraic perspective agree with the existing ones.
