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The correspondence between definable connected groupoids in a theory $T$ andinternal 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 correspondenceis in fact an equivalence of categories, with respect to appropriate notions ofmorphism. Secondly, the equivalence of categories is shown to vary uniformly indefinable families, with respect to an appropriate relativisation of thesecategories. Some elaboration on Hrushovki's original constructions are alsoincluded.

Functoriality and uniformity in Hrushovski's groupoid-cover correspondence