Consider two copies of holomorphic disks $\{ z \in \mathbb{C} \ | \ |z| \leq 1 \}$. Denote them by $\Delta_1$ and $\Delta_2$. Let $f$ be a diffeomorphism from boundary of first disk to boundary of the second one $f: \partial \Delta_1 \rightarrow \partial \Delta_2$.
If we glue $\Delta_1$ and $\Delta_2$ by $f$ then we will get a sphere. Require that $f$ compatible with orientation (orientations, coming from complex structure, can be glued to orientation of sphere).
Question: Can we get a comlex structure of sphere such that it's restriction on interior of each disk is initial complex structure?
The answer is yes. This is the so-called conformal welding problem, which can be solved for diffeomorphisms. See my answer there.