$C^\infty$-differential structures of $\mathbb{R}$ up to $C^\infty$-diffeomorphism

Question

I want to show that

The $C^\infty$-differential structures of $\mathbb{R}$ up to $C^\infty$-diffeomorphism is unique.

For differential structures given by a single chart $(\varphi,\mathbb{R})$, I think the diffeomorphism to the differential structure given by $(\mathrm{id},\mathbb{R})$ can be found by taking its inverse.

But I don't know how to find the diffeomorphism for a differential structure given by multiple charts, say $(\varphi_1,U_1)$, $(\varphi_2,U_2)$. I can take their inverse repectively, but what should I do on the intersections?

Thanks in advance!