Let us consider the square $I \times I$ and define on its boundary the following equivalence relation: $(0,t) \sim (1,t)$ for any $t \in I$ and $(\frac{1}{2},0) \sim (\frac{1}{2},1)$. Which is the fundamental group of $X=\partial (I \times I)/\sim$?

I argue as follows: $X$ is a cylinder with two points identified (the centers of the circular bases). I can consider two arcs $A$ and $B$ joining the previous points but $B$ all contained in the cylinder and $A$ intersecting the cyilinder only at the endpoints. In this way, it should be $\pi\_1(X)=\pi\_1((S^1 \times \mathbb{R}) \vee S^1)=\mathbb{Z} \* \mathbb{Z}$. Is my argument right or not?