I am constructing a degree $2$ map from the genus-two surface $S_2$ to the genus-one surface $S_1$. Searching on this website, I noticed the following approach: Let $\Sigma$ be a compact connected bordered subsurface of $S_2$ such that $\Sigma$ has genus one and has exactly one boundary component. Then the quotient space $S_2/\Sigma$ obtained by identifying $\Sigma$ to a point is homeomorphic to $S_1$. Consider the map $f\colon S_2\to S_1$ obtained by identifying $\Sigma$ to a point. Then $\deg (f)=\pm 1$. Note that $\varphi_n\colon\Bbb S^1\times \Bbb S^1\ni (z,w)\longmapsto (z^n,w)\in \Bbb S^1\times \Bbb S^1$ is a self-map of $S_1$ with degree $n$. For suitable $n$, the map $\varphi_n\circ f\colon S_2\to S_1$ is of degree $2$ as degree is multiplicative.
I am trying to construct a degree two map $S_2\to S_1$ in a different way which is given below. Can someone tell me whether the following approach is correct or not?
Edit after Ted Shifrin's comment: Consider the map $g\colon\Bbb C\ni z\longmapsto z^2\in\Bbb C$. Now, $g$ is a proper map and $g\big|\Bbb R^2\backslash \mathbf 0\to\Bbb R^2\backslash \mathbf 0$ is a two-fold cover. So, considering induced orientation on the domain, we can say $g\big|\Bbb R^2\backslash \mathbf 0\to\Bbb R^2\backslash \mathbf 0$ is of degree two. All these orientations can be extended to the whole manifolds, and then the branched covering $g$ can be thought of as a map of degree two. Since $g$ is proper, we can extend it to map $G\colon \Bbb S^2\to \Bbb S^2$. Note that $\deg(G)=2$
Let $B$ be a subset of $\Bbb R^2$ homeomorphic to the closed unit disk, such that $g^{-1}(B)=B_1\sqcup B_1$ where each $g\big|B_i\to B$ is an orientation preserving homeomorphism.
Now, $G$ restricted to $\Bbb S^2\big\backslash \big(\text{int}(B_1)\cup\text{int}(B_2)\big)\to \Bbb S^2\big\backslash\text{int}(B)$ is also a degree two map. Note that $S_2$ is obtained by adding two handles to $\Bbb S^2\big\backslash \big(\text{int}(B_1)\cup\text{int}(B_2)\big)$ via a orientation-reversing map $\partial B_1\sqcup \partial B_2\to \partial(\Sigma\sqcup\Sigma)$, where $\Sigma$ is the compact connected bordered surface of genus one with one boundary component. Similarly, $S_1$ is obtained by adding a handle to $\Bbb S^2\big\backslash\text{int}(B)$. Now, one can extend $G$ over the handles to get a map $\widetilde G\colon S_2\to S_1$ with $\deg\left(\widetilde G\right)=2$. So, we are done.