  We extend the theory of relative trisections of smooth, compact, oriented
$4$-manifolds with connected boundary given by Gay and Kirby to include
$4$-manifolds with an arbitrary number of boundary components. Additionally, we
provide sufficient conditions under which relatively trisected $4$-manifolds
can be glued to one another along diffeomorphic boundary components so as to
induce a trisected manifold. These two results allow us to define a category
$\textrm{Tri}$ whose objects are smooth, closed, oriented $3$-manifolds
equipped with open book decompositions, and morphisms are relatively trisected
cobordisms. Additionally, we extend the Hopf stabilization of open book
decompositions to a relative stabilization of relative trisections.
