In section 2.3 on p. 16 of the book "Basic Concepts of String Theory" by Blumenhagen, Lüst, Theisen, 3 symmetries of Polyakov action are discussed: Poincarè invariance, diffeomorphism invariance and weyl symmetry. In attempt to simplify the action, the authors make use of the conformal gauge such that in page 16 we have the following:
The local invariances allow for a convenient gauge choice for the world-sheet metric $h_{\alpha \beta}$, called conformal or orthonormal gauge. Reparametrization invariance is used to choose coordinates such that locally $h_{\alpha \beta}=\Omega^{2}(\sigma, \tau) \eta_{\alpha \beta}$ with $\eta_{\alpha \beta}$ being the two-dimensional Minkowski metric defined by $d s^{2}=-d \tau^{2}+d \sigma^{2} .$ It is not hard to show that this can always be done. Indeed, for any two-dimensional Lorentzian metric $h_{\alpha \beta}$, consider two null vectors at each point. In this way we get two vector fields and their integral curves which we label by $\sigma^{+}$ and $\sigma^{-}$. Then $d s^{2}=-\Omega^{2} d \sigma^{+} d \sigma^{-} ; h_{++}=h_{--}=0$ since the curves are null. Now let \begin{equation} \sigma^{\pm} = \tau \pm \sigma\tag{2.34} \end{equation} from which it follows that $d s^{2}=\Omega^{2}\left(-d \tau^{2}+d \sigma^{2}\right)$. A choice of coordinate system in which the two-dimensional metric is conformally flat, i.e. in which $$ d s^{2}=\Omega^{2}\left(-d \tau^{2}+d \sigma^{2}\right)=-\Omega^{2} d \sigma^{+} d \sigma^{-}\tag{2.35} $$is called a conformal gauge. The world-sheet coordinates $\sigma^{\pm}$ introduced above are called light-cone, isothermal or conformal coordinates. In these coordinates $\gamma_{\alpha \beta} \equiv \frac{h_{\alpha \beta}}{\sqrt{-h}}=\eta_{\alpha \beta} .$ We can now use Weyl invariance to set $h_{\alpha \beta}=\eta_{\alpha \beta}$
My problem is: the authors make use of the conditions $$h_{++}d\sigma^+d\sigma^+ = h_{--}d\sigma^{-}d\sigma^- = 0 \implies h_{++}=h_{--}=0,$$ since $\sigma^{\pm}$ are diffeomorphisms by assumpiton. After this, the author choose an explicit form for those diffeormorphism, using the so called light-cone coordinates. Doesn't making the functions explicit imply the use of the two reparametrization symmetries? If so, the condition stablished that $h_{++}=h_{--}=0$ wouldn't need an "extra symmetry"?
