My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following.
Let $X$ be a topological space, and $U, V\subseteq X$ two path-connected open subsets such that $U\cap V\subseteq X$ is path connected. Consider the push-out of the commutative diagram
$ \qquad \qquad \qquad \qquad \qquad \qquad \qquad\pi_{1}(U)\longleftarrow \pi_{1}(U\cap V) \longrightarrow \pi_{1}(V) $
in the category of groupoids. Then this colimit is isomorphic to $\pi_{1}(X)$.
I would like to find a simple proof of this statement, possibly one that is constructed by verifying directly the universal property on $\pi_{1}(X)$. Links to (free) further readings are also welcome - I found a paper that gives such description a long time ago, but I cannot find it anymore. In what way is the hypothesis that the open subsets $U$, $V$ and $U\cap V$ are path-connected useful? Is there a version of this statement that uses weaker hypothesis on the covering?