Differential geometry and transition maps as communative diagrams

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Consider the picture below.enter image description here

I know pictures are usually looked down upon on this site but I have no experience with TikZ.

Anyway, I am learning about regular surfaces and transition maps in differential geometry. We have some regular surface $S$ and open sets $\mathcal{U}$ and $\mathcal{\tilde{U}}.$

Right now, I am trying to make a communative diagram with all of these maps to understand this better. But, I have never made a communative diagram in my life. Is anyone able to help me? And, is this information enough?

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