In 2015, Christopher Daw and Adam Harris proved that Shimura curves are categorical. Roughly, this means that if $p:X^{+}\rightarrow S(\mathbb{C})$ denotes a Shimura curve (here $X^{+}$ is the Hermitian symmetric domain of $S$), then $p$ is, up to some kind of automorphism, uniquely determined by the algebraic equations that it satisfies and the size of its fibres. I will explain how one can extend their method to higher dimensional Shimura varieties and give an analogous categoricity result which is conditional upon a conjecture of Pink.