Let $D_1$ and $D_2$ be a Jordan domains on $\mathbb{S}^2$. More formally, there exist two continuos injective maps $\phi$ and $\psi$ from $B = \{z \in \mathbb{C}||z| \leq 1\}$ to $\mathbb{S}^2$ and $D_1 = \phi(B),\, D_2 = \psi(B)$. I want to prove that there exists homeomorphism $f$ of $\mathbb{S}^2$ such that $f(D_1) = D_2$ and $f$ is isotopic to identity.
My idea was to extend $\phi$ and $\psi$ to homeomorphisms $\hat{\phi}$ and $\hat{\psi}$ of $\mathbb{S}^2$, which are homotopic to each other, then use the result that any two homotopic homeomorphisms of a compact surface(unless this surface is closed annulus or closed disk) are actually isotopic. This theorem(Theorem 1.12, page 41) is stated in A Primer on Mapping Class Groups by Benson Farb and Dan Margalit.
To construct such nice extensions I was wondering to apply the Jordan-Schonflies theorem, which says that $\partial D_1$ and $\partial D_2$ nicely separate $\mathbb{S}^2$. For homotopy equivalence, I was thinking about using the fact that if a continuous map $h:\mathbb{S}^1 \to X$ extends to a continuous map $k: B \to X$ then $h$ is nullhomotopic(for example, see Lemma 55.3, page 349 from Topology, Second Edition, by James R. Munkres).
Unfortunately, I still not able to mix all the ideas in order to obtain complete proof.
P.S. Fact that for any two Jordan domains $X, Y$ of connected surface $M$ there exists homeomorphism of $M$ isotopic to identity and mapping $X$ to $Y$ is stated on page 422 in Expanding Thurston Maps by Mario Bonk and Daniel Meyer, https://arxiv.org/abs/1009.3647, also here is an outline of a proof. Nevertheless, this proof sounds to me not complete enough, also I am particularly interested in the case of the 2-sphere and in completing my own arguments.