My definitions are as follows. $\hat{\mathbb{C}}$ is the Riemann sphere. Degree of a continuous map $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ is how many times it wraps around $\hat{\mathbb{C}}$. Post critical set of $f$ is $P_{f}:=\bigcup_{i>0} f^{\circ i}\left(C_{f}\right)$, where $C_{f}$ is the set of critical points of $f$. Thurston function is an orientation-preserving branched covering of degree $\geq2$ and its post critical set is finite. Quadratic means degree 2.
I am guessing that due to the Riemann-Hurwitz Theorem a quadratic Thurston function $f$ has $2d-2=2.2-2=2$ critical points (counting multiplicities). $f$ fails to be injective in any neighborhood of a critical point. due to the definition of a critical point. If I suppose that two critical points are the same, pictorially around a neighborhood of the critical point I have 4 points mapping to a critical value and that is a contradiction to map being quadratic.
Is the statement in the title correct?
Is the fact that a quadratic Thurston function has two critical points counting multiplicities correct?
Is my proof correct, if so how can I improve my argument?
Do I need the extra assumption that $f$ is rational, that is ratio of two polynomials?
Any help is appreciated.