We have the fact that for any path connected space, its fundamental group is abelian iff the isomorphism between the fundemental groups (of different base point) doesn't depend on the choice of the path. (See this question) My question is, is there some categorical expression of this proposition, like something involving natural isomorphism of functors?
The idea comes from the sentence like "the isomorphism doesn't depend on the choice of xxx". For example, we say that for any finite dimension vector space $V$, the isomorphism between $V$ and its double dual $V^{**} $ doesn't depend on the choice of the basis in $V$ (which is not the case for $V$ and $V^*$). For this we have a statement like "the identity functor and the double dual functor is naturally isomorphic".
The main difficulty seems to be that, unlike the example of the vector space (where we choose $V$ and get two objects, $V$ and $V^{**} $), we first choose one space $X$ and then two different point $x_0$, $x_1$ for two objects $π_1(X, x_1)$, $π_1(X, x_2)$. Thus I don't know how to construct two functors. Is there some way to get through this, or we actually need something more than functors and natural isomorphism?