  $\mathbf{Theorem.}$
  Let $K$ be an abstract elementary class (AEC) with amalgamation and no
maximal models. Let $\lambda > \text{LS} (K)$. If $K$ is categorical in
$\lambda$, then the model of cardinality $\lambda$ is Galois-saturated.
  This answers a question asked independently by Baldwin and Shelah. We deduce
several corollaries: $K$ has a unique limit model in each cardinal below
$\lambda$, (when $\lambda$ is big-enough) $K$ is weakly tame below $\lambda$,
and the thresholds of several existing categoricity transfers can be improved.
  We also prove a downward transfer of solvability (a version of superstability
introduced by Shelah):
  $\mathbf{Corollary.}$
  Let $K$ be an AEC with amalgamation and no maximal models. Let $\lambda > \mu
> \text{LS} (K)$. If $K$ is solvable in $\lambda$, then $K$ is solvable in
$\mu$.
