  In arXiv:2011.03313, the author has constructed a category of abstractly
automorphic representations for $\mathrm{GL}(2)$ over a function field $F$.
This is a symmetric monoidal Abelian category, constructed with the goal of
having the irreducible automorphic representations as its simple objects. The
goal of this paper is to systematically study this category.
  We will prove several structural theorems about this category. We will show
that it admits an adjoint pair $(r^\mathrm{aut},i^\mathrm{aut})$ of automorphic
parabolic restriction and induction functors, respectively. This will allow us
to show that the category of abstractly automorphic representations decomposes
into cuspidal and Eisenstein components, in analogy with the Bernstein
decomposition of the category of $p$-adic representations.
  Moreover, along the way, we will give a new perspective on the intertwining
operator of $\mathrm{GL}(2)$ (and on the functional equation for Eisenstein
series), as a form of self-duality of the functor of parabolic induction. We
will also illustrate how the role of analytic continuation in this theory can
be thought of as trivializing a twist by a certain line bundle, which
corresponds to an L-function via the results of arXiv:2012.03068. If one
chooses to keep the twist as a part of the theory, then one avoids the need for
analytic continuation.
