  We use the theta correspondence to study the equivalence between
Godement-Jacquet and Jacquet-Langlands L-functions for $\mathrm{GL}(2)$. We
show that the resulting comparison is in fact an exotic symmetric monoidal
structure on the category of $\mathrm{GL}(2)$-modules.
  Moreover, this enables us to construct an Abelian category of abstractly
automorphic representations, whose irreducible objects are the usual
automorphic representations. We speculate that this category is a natural
setting for the study of automorphic phenomena for $\mathrm{GL}(2)$, and
demonstrate its basic properties.
  This paper is a part of the author's thesis.
