  We prove that a universal class categorical in a high-enough cardinal is
categorical on a tail of cardinals. As opposed to other results in the
literature, we work in ZFC, do not require the categoricity cardinal to be a
successor, do not assume amalgamation, and do not use large cardinals. Moreover
we give an explicit bound on the "high-enough" threshold:
  $\mathbf{Theorem}$
  Let $\psi$ be a universal $\mathbb{L}_{\omega_1, \omega}$ sentence. If $\psi$
is categorical in some $\lambda \ge \beth_{\beth_{\omega_1}}$, then $\psi$ is
categorical in all $\lambda' \ge \beth_{\beth_{\omega_1}}$.
  As a byproduct of the proof, we show that a conjecture of Grossberg holds in
universal classes:
  $\mathbf{Corollary}$
  Let $\psi$ be a universal $\mathbb{L}_{\omega_1, \omega}$ sentence that is
categorical in some $\lambda \ge \beth_{\beth_{\omega_1}}$, then the class of
models of $\psi$ has the amalgamation property for models of size at least
$\beth_{\beth_{\omega_1}}$.
  We also establish generalizations of these two results to uncountable
languages. As part of the argument, we develop machinery to transfer
model-theoretic properties between two different classes satisfying a
compatibility condition. This is used as a bridge between Shelah's milestone
study of universal classes (which we use extensively) and a categoricity
transfer theorem of the author for abstract elementary classes that have
amalgamation, are tame, and have primes over sets of the form $M \cup \{a\}$.
