In finding the fundamental group of the torus by using the Van Kampen theorem, I was reading this: "Let us choose the covering consisting of the punctured torus U, an open disk V that covers the puncture, and their intersection (a punctured disk).... $\pi _1 (V) =0$ and $\pi _1 (U)$is free group of two generators $\alpha ,\beta $..... Thus, $\pi _1(U\cap V)$ is infinite cyclic on one generator $\gamma$. Visualizing the punctured torus once again as its fundamental polygon, any lift of $\phi _1(\gamma )$ is a loop around the puncture (where $\phi _1:\pi _1(U\cap V)\rightarrow \pi _1 (U)$ ). Deforming this loop to the edge of the square, it can be seen that$\phi _1(\gamma )=\alpha \beta \alpha^{-1} \beta ^{-1}$ "
I need to understand the last sentence, I mean why $\phi _1(\gamma )=\alpha \beta \alpha^{-1} \beta ^{-1}$?
here is the link which i was reading from http://people.reed.edu/~jerry/332/projects/colley.pdf page 4