It is a well known fact that if $X$ is a topological space where there exist a path with endpoints $x,y\in X$ then the fundamental group of the pointed spaces $(X,x)$ and $(X,y)$ are isomorphic, though not 'canonically', in the sense that the isomoprhism is directly induced by the path mattered.
As a simple consequence, if $X$ is path connected the fact holds for any choice of $x,y\in X$.
I was wondering if we can translate this result in a statement about the fundamental group functor. In fact, the path induces an isomorphism between groups, anyhow the path itself is not a morphism of the category of pointed topological spaces. So intuitively there is a functorial flavour, which I do not if can be made precise.
Moreover one can prove that the isomorphism is path independent hence 'canonical' if and only if the fundamental group is abelian. Also in this case, I wonder if there is a nice categorical intepretation.