About the automorphism groups of the objects in a connected groupoid In the note named Foundation of Algebraic Geometry, the author gives an example: given a topological space $X$, there is a fundamental groupoid which is the category in which the objects are points of $X$, and morphisms $x\rightarrow y$are paths from $x$ to $y$ up to homotopy. We see the automorphism group $Aut(x)=\pi_1(X,x)$ is the fundamental group.
Then, the author points out that

In the case where $X$ is connected, and $\pi_1(X)$ is not abelian, this illustrates the fact that for a connected groupoid, the automorphism groups of objects are all isomorphic, but not canonically isomorphic.

I don't understand this words. I see that in a connected groupoid, all objects are isomorphic to each other, thus their automorphism groups are all isomorphic. But why aren't they "canonically isomorphic"? What does it mean?

It means that while an isomorphism between fundamental groups at various points always exists, it's not unique if the fundamental group is non-abelian. To see this, consider two paths $h,h'$ from $x$ to $y$.
Note that $\gamma\mapsto h\gamma h^{-1}$ and $\gamma\mapsto h'\gamma h'^{-1}$ are both isomorphisms $\pi_1(X,x)\to \pi_1(X,y)$, but they are distinct unless $\gamma_0=h^{-1}h'$ is in the center of  $\pi_1(X,x)$ (as you can write the latter as $\gamma\mapsto (h\gamma_0)\gamma (h\gamma_0)^{-1}=h(\gamma_0\gamma\gamma_0^{-1})h^{-1}$). So in particular, if $\pi_1(X,x)$ is abelian, they are always the same, and conversely, if any the isomorphism is unique, the group must be abelian.
