Categoricity for transfinite extensions of modules

For each deconstructible class of modules $\mathcal{D}$, we prove that the categoricity of $\mathcal{D}$ in a big cardinal is equivalent to its categoricity in a tail of cardinals. We also prove Shelah’s Categoricity Conjecture for $(\mathcal{D},\prec )$, where $(\mathcal{D},\prec )$ is any abstract elementary class of roots of Ext.