On my complex analysis prelim this morning I was asked to give a conformal map from the region $L=\{z\in\mathbb{C}:|z-i|<\sqrt{2},|z+i|<\sqrt{2}\}$, a lune with vertices at $-1$ and $1$ to the unit disc $\mathbb{D}=\{z:|z|<1\}$. I tried to send $L$ to the upper half plane by the Möbius transform sending $(-1,0,1)$ to $(0,i,\infty)$. Then I composed with the Cayley transformation to get to the unit disc.
My question is: does my first map do what I want it to(presuming I calculated it correctly)?
To be brief, does the Möbius transform which takes $(-1,0,1)$ to $(0,i,\infty)$ send $L$ to the upper half plane?