Problem 5-1 from John Lee's Introduction to Smooth Manifolds ask us to show that the submanifold $\Phi^{-1}(0,1) \subset \mathbb{R}^4$ is diffeomorphic to $\mathbb{S}^2$, where $\Phi: \mathbb{R}^4 \to \mathbb{R}^2$ is given by $\Phi(x,y,s,t) = (x^2 + y, x^2 + y^2 + s^2 + t^2 + y)$.
For me, it seems natural to try to show the map $F: \Phi^{-1}(0,1) \to \mathbb{S}^2$ given by $F(x,y,s,t) = (y,s,t)$ is a diffeomorphism. However the condition $x^2 + y = 0$ implies that if $(x,y,s,t) \in \Phi^{-1}(0,1)$, then $y \leq 0$ and so the map $F$ cannot be surjective.
Any idea is welcome!
You are right in that your suggested map $F(x,y,s,t)=(y,s,t)$ is not surjective. To get an idea how to fix that, notice that it is also not injective, because $F(x,y,s,t)=F(-x,y,s,t)$.
The correct diffeomorphism should be $F(x,y,s,t)=(sign(x) \cdot y,s,t)$. It solves both your problems of surjectivity and injectivity. It remains to check that the map is smooth for points with $x=0$.