Consider the affine scheme $$ X = \text{Spec}\left( \frac{\mathbb{C}[t,x]}{(x^2 - t)} \right) $$ If I take the projection onto the $t$-axis, there is an obvious $\mathbb{Z}/2$-action switching the fibers. This can be described by sending $x \mapsto -x$. How can I find the hopf-algebra action (maybe co-action) of $\text{Spec}(\mathbb{C}[s]/(s^2 - 1)) = \mathbb{Z}/2$ on the algebra defining $X$? I want to try and get an explicit diagram of the cogroupoid of algebras.
My first guess is the action (co-action?) sends $$ x \mapsto -x\otimes s \text{ and } t \mapsto t\otimes s $$ for the target map $$ \frac{\mathbb{C}[t,x]}{(x^2 - t)} \to \frac{\mathbb{C}[t,x]}{(x^2 - t)}\otimes_\mathbb{C} \frac{\mathbb{C}[s]}{(s^2 - 1)} $$ and $$ x\mapsto x \text{ and } t\mapsto t $$ for the source map.
Also, I think the identity section should be the map $$ \frac{\mathbb{C}[t,x]}{(x^2 - t)}\otimes_\mathbb{C} \frac{\mathbb{C}[s]}{(s^2 - 1)} \to \frac{\mathbb{C}[t,x]}{(x^2 - t)} $$ sending $$ s\to 1 $$