In Butz, Moerdijk, Representing topoi by topological groupoids, Section 4, it is said that in order to prove that the fibres in a point $p$ of the source (domain) and target (codomain) maps (respectively $s$ and $t$) are both homeomorphic to some space $En(S_p)$, it suffices to prove it just for the source map.
Why is this true? Does it come from some form of abstract relationship (homeomorphism?) between $s^{-1}(p)$ and $t^{-1}(p)$ that I can't see?
Thank you in advance.
Yes, they are homeomorphic by the inverse map $i:G\to G$. Since $i$ interchanges sources and targets, it maps $s^{-1}(p)$ to $t^{-1}(p)$ (and is a homeomorphism because it is its own inverse).