Doubt in passing in the Riemann mapping theorem 
I have a question, maybe silly in the passage marked in red. I understood everything up to this part. Why does $H'(0) > 0$ imply $e^{i \theta} = 1$? Is the Schwarz Lemma being used?

From $H=F \circ G^{-1}$ we get with the chain rule and the rule for the derivative of $G^{-1}$ that
$$ H'(0)= \frac{F'(z_0)}{G'(z_0)}.$$
Since $F'(z_0), G'(z_0)>0$, it follows that $H'(0)>0$.
From $H(z)=e^{i \theta}z$,  we get $H'(0)=e^{i \theta}.$
Furthermore we have: $e^{i \theta}>0 \iff e^{i \theta}=1.$
