  Motivated by the free products of groups, the direct sums of modules, and
Shelah's $(\lambda,2)$-goodness, we study strong amalgamation properties in
Abstract Elementary Classes. Such a notion of amalgamation consists of a
selection of certain amalgams for every triple $M_0\leq M_1, M_2$, and we show
that if $K$ designates a unique strong amalgam to every triple $M_0\leq M_1,
M_2$, then $K$ satisfies categoricity transfer at cardinals
$\geq\theta(K)+2^{\text{LS}(K)}$, where $\theta(K)$ is a cardinal associated
with the notion of amalgamation. We also show that if such a unique choice does
not exist, then there is some model $M\in K$ having $2^{|M|}$ many extensions
which cannot be embedded in each other over $M$. Thus, for AECs which admit a
notion of amalgamation, the property of having unique amalgams is a dichotomy
property in the sense of Shelah's classification theory.
