How do I find all group homomorphisms, where $\psi$ is continuous for the Zariski topology: $$\psi:\left\{\begin{bmatrix}a&b\\0&a^{-1}\end{bmatrix}:a\in \Bbb C^\times,b\in \Bbb C\right\}\to \left\{\begin{bmatrix}c&0\\0&c^{-1}\end{bmatrix}:c\in \Bbb C^\times\right\}$$
I tried to define the image matrix entries as functions of the entries of the domain matrices, and I tried to use the fact that inverses go to inverses, but couldn't make any progress.
$$\psi(\begin{bmatrix}a&b\\0&a^{-1}\end{bmatrix})=\begin{bmatrix}f_1(a,b)&0\\0&f_1(a,b)^{-1}\end{bmatrix}$$
I saw that if we treat just the diagonal of the domain matrices, we see that $$\psi(\begin{bmatrix}a^n&0\\0&a^{-n}\end{bmatrix})=\begin{bmatrix}f_1(a^n)&0\\0&f_1(a^n)^{-1}\end{bmatrix}=\begin{bmatrix}f_1(a)^n&0\\0&f_1(a)^{-n}\end{bmatrix}$$
Which seems like now I would need to analyse $f_1:\Bbb C^\times \to \Bbb C^\times$?
I haven't yet tried to use continuity as seen in the comments. Like $$\psi:[\Bbb A^1\times (\Bbb A^1 -\{0\})]\to \Bbb A^1-\{0\}.$$