Title: Functoriality and uniformity in Hrushovski's groupoid-cover correspondence,
Abstract: The correspondence between definable connected groupoids in a theory $T$ and
internal generalised imaginary sorts of $T$, established by Hrushovski in
["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics,
2012], is here extended in two ways: First, it is shown that the correspondence
is in fact an equivalence of categories, with respect to appropriate notions of
morphism. Secondly, the equivalence of categories is shown to vary uniformly in
definable families, with respect to an appropriate relativisation of these
categories. Some elaboration on Hrushovki's original constructions are also
included.