The concept is indeed a bit subtle. Let’s restate the definition of $\phi$, which is supposed to take an $H$-coset and assign to it a $K$-coset. The definition can be read this way: “Given an $H$-coset $S$, $\phi(S)$ is defined to be a $K$-coset as follows. Choose any element $a$ of $S$, and then $\phi(S)$ will be the $K$-coset that contains $a$.”
Now it becomes clear that the resulting $K$-coset apparently depends on your choice of $a$: if you chose a different element of $S$, would you get the same $K$-coset by applying the recipe to that? Showing that which element of $S$ you chose doesn’t affect the final resulting $K$-coset is just what is being asked for.