= Solution
Regard the <3-sphere> as
$$
S^3=\{(z_1,z_2)\in\mathbb C^2:|z_1|^2+|z_2|^2=1\}.
$$
It is the union of the two closed solid tori
$$
U'=\{|z_1|\geq|z_2|\},
\qquad
V'=\{|z_2|\geq|z_1|\}.
$$
Explicit homeomorphisms from the two copies in the question are
$$
\Phi_U(u,w)=\frac{(u,w)}{\sqrt{1+|w|^2}},
\qquad (u,w)\in S^1\times D^2,
$$
and
$$
\Phi_V(z,v)=\frac{(z,v)}{\sqrt{1+|z|^2}},
\qquad (z,v)\in D^2\times S^1.
$$
On their boundaries both maps send $(u,v)\in S^1\times S^1$ to $(u,v)/\sqrt2$. They therefore descend through the quotient to a continuous bijection $S\to S^3$. Since $S$ is compact and $S^3$ is Hausdorff, this bijection is a homeomorphism.
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