Morita equivalence of Lie groupoids and isomorphism of differentiable stacks

Question

It's a well known fact two Lie groupoids are Morita-equivalent iff they induce isomorphic differentiable stacks (I'll call this statement "(1)").

It's also well known that there is a biequivalence between the bicategory of Lie Groupoids with right principal bibundles and the strict 2-category of differentiable stacks (I'll call this statement "(2)").

I've always thought that the $(2)\Rightarrow (1)$, but now I realized that biequivalences only preserve equivalences so the fact (2) just implies two Lie groupoids are Morita-equivalent iff their induced differentiable stacks are equivalent (not isomorphic).

How do I fix this?

Answer 1

The "well-known fact" is simply not true if you assume "isomorphic stacks" means literally isomorphic (say as fibred categories). My impression is that people who work in certain parts of algebraic geometry say "isomorphic" here when they really mean "equivalent". The correct statement is as you indicate: two Lie groupoids are Morita equivalent (i.e. equivalent in the bicategory of Lie groupoids and principal bibundles, or in the bicategory of Lie groupoids and anafunctors) if and only if their associated stacks are equivalent as fibred categories (which is the same, modulo foundational fuzz that is almost not surely relevant to you, as the corresponding pseudofunctors to $\mathbf{Gpd}$ being naturally equivalent).