I'm preparing a presentation where I'd like to present a proof of Picard's little theorem using hyperbolic geometry. Picard's little theorem states that the range of an entire function can omit at most $1$ point.
I can see that we can write $U:=\mathbb{C}P^1\setminus\{\infty,0,1\}$ as a hyperbolic manifold, and as a quotient of $\mathbb{D}$ through the Farey tesselation. I would like to say therefore that any conformal map $:\mathbb{C}\rightarrow U$ would factor conformally through $\mathbb{D}$.
I can see that it should factor through the covering map, but can I even say that the covering map is conformal?
I know that Ahlfors' Lemma helps, but all the statements I've obtained of said lemma hold for Hermite metrics. Can I say that the hyperbolic metric on $\mathbb{C}P^1\setminus\{\infty,0,1\}$ is Hermite?