Check that complex charts being compatible

Question

I'm reading Algebraic Curves and Riemann Surfaces by Rick Miranda. I let $X = \mathbb{R}^2$ and let $U$ be any open subset of $X$. Then I define $\phi_{U} (x,y) = x+iy$ from $U$ to $\mathbb{C}$, being a complex chart on $\mathbb{R}^2$. Then it wants me to prove that any two charts defined like this are compatible with each other.
Then I first try to use Cauchy-Riemann Equation. I have $\frac{\partial u}{\partial x} = 1$, $\frac{\partial u}{\partial y} = 0$, $\frac{\partial v}{x} = 0$, and $\frac{\partial v}{\partial y} = 1$ And this means that the chart satisfies Cauchy-Riemann Equation, and since all four partial derivatives are continuous, then this means that $\phi_{U}(x,y)$ is holomorphic for every $(x,y) \in \mathbb{R}^2$.
Then on the book, the definition of being compatible is that we let $\phi_{1}: U_{1} \rightarrow V_{1}$ and $\phi_{2}: U_{2} \rightarrow V_{2}$ be two complex charts on $X$. We say that $\phi_{1}$ and $\phi_{2}$ are compatible if either $U_{1} \cap U_{2} = \emptyset$, or $\phi_{2} \circ \phi_{1}^{-1}: \phi_{1} (U_{1} \cap U_{2}) \rightarrow \phi_{2} (U_{1} \cap U_{2})$ is holomorphic. But I'm confused how I should pick my $phi_{1}$ and $\phi_{2}$ in my specific context.
UPDATE: Here my $\phi_{1}$ and $\phi_{2}$ will be the same charts. Then I have $$\phi_{2} \circ \phi_{1}^{-1} = x+iy$$ which is holomorphic, since it's just a linear map.