I was reading Lectures on Moduli Spaces of Elliptic Curves https://arxiv.org/abs/0812.1803, and stuck at Exercise 20: Show that the fundamental group of a basic orbifold is a group.
I realized that a "pointed orbifold map" $(f,\phi)$ from $S^1 = (\mathbb{R}, \mathbb{Z}, +n)$ is fully determined by $f|_{[0,1]}$ and $\phi(1)$.
Suppose I have two maps $(f,\phi)$ and $(f',\phi')$. Let $g = \phi(1), h = \phi'(1)$.
How could I define their composition as ordinary fundamental group? If $f(1)$ and $f'(0)$ are in the same conjugacy class, let $u f(1) = f'(0)$, I can produce a new path $u f|_{[0,1]}$ to compose with $f'$.
But $f(1)$ and $f'(0)$ could be in different conjugacy classes, could I find an orbifold homotopy to solve this?