The fundamental groupoid $\Pi_1(X)$ is the category whose objects are points of $X$ and whose morphisms are endpoint-preserving homotopy classes of paths in $X$, with composition defined via the gluing of the unit interval just described. The reason that morphisms are endpoint-preserving homotopy classes, rather than simply paths, is that path composition is not strictly associative or unital. Endpoint-preserving homotopies allow in particular for re-parameterizations of the unit interval.