  We provide a complete classification of all the possible categoricity
spectra, in terms of internal size, that can appear in a large accessible
category with directed colimits, assuming the Singular Cardinal Hypothesis
($SCH$), and providing as well explicit threshold cardinals for eventual
categoricity. This includes as a particular case the first complete
classification of categoricity spectra of abstract elementary classes (AEC's)
entirely in $ZFC$. More specifically, we have the following theorem:
  Let $\mathcal{K}$ be a large $\kappa$-accessible category with directed
colimits. Assume the Singular Cardinal Hypothesis $SCH$ (only if the
restriction to monomorphisms is not an AEC). Then the categoricity spectrum
$\mathcal{C}at(\mathcal{K})=\{\lambda\geq \kappa: \mathcal{K} \text{ is
$\lambda$-categorical}\}$ is one of the following:
  1) $\mathcal{C}at(\mathcal{K})=\emptyset$.
  2) $\mathcal{C}at(\mathcal{K})=[\alpha, \beta]$ for some $\alpha, \beta \in
[\kappa, \beth_{\omega}(\kappa))$.
  3) $\mathcal{C}at(\mathcal{K})=[\chi, \infty)$ for some $\chi \in [\kappa,
\beth_{(2^{\kappa})^+})$.
  This solves in particular Shelah categoricity conjecture for AEC's. There are
examples of each of the three cases of the classification, showing that they
indeed occur.
