Suppose we have some meromorphic quadratic differential $q=\phi\left(x\right)\mathrm{d}x^2$ on a punctured Riemann surface $\mathcal{C}$, and that $q$ is the pullback via a function $f$ of a meromorphic quadratic differential with three poles on the sphere $\mathbb{P}^1$:
$$q=f^*\left(\frac{\left(\mathrm{d}\zeta\right)^2}{4\pi^2\zeta\left(1-\zeta\right)}\right)$$
(For reference, I am looking at Theorem 6.5 of this paper).
My question is: just by using the definition of the pullback and the fact that $f^*\left(\mathrm{d}g\right)=\mathrm{d}\left(f^*g\right)$, can we then write:
$$q=\frac{\left(\mathrm{d}f\right)^2}{4\pi^2f\left(1-f\right)}$$
This seems innocuous from the definition of the pullback, but I am suspicious for two reasons: (a) I see the first equation in the literature a lot, but never the second; (b) something fishy happens in my subsequent calculations if I push through with this.
Should be a simple question to answer, but any help much appreciated!
The answer to this question is that yes, the second expression for $q$ above is fine. This expression follows from the definition of the pullback, and the problems I was encountering in later calculations were due to other issues.