I would like to ask for help understanding May's (concise course of algebraic topology) proof of the Van Kampen theorem through colimits. Explicitly, I don't understand how to construct the inverse equivalence $F:\Pi(X) \to \pi_1(X,x)$ in such a way that $FJ=Id$, when $J:\pi_1(X,x)\to \Pi(X) $ is the inclusion. In the book says that one should make a choice of path classes $x\to y$ and whenever $y=x$ one should take $c_x$. I think that this last choise does not make sense, for it losses all the information of the classes in $\pi_1(X,x)$. Certainly, for $FJ$ the function on objects is the identity but the function on the morphisms is constant, $c_x$.
Even more, for the case $x\neq y$ the choise of classes is more troublesome if one wants to respect the composition of paths in $\Pi(X)$ and the group structure in $\pi_1(X,x)$.
Any help would be appreciated.