I've covered the tangent and cotangent bundles in my previous article Vectors and Covectors, which you can find in the link below:

https://joseph-mellor1999.medium.com/vectors-and-covectors-81b64ff6e38b?source=friends_link&sk=da7b102f327edf03cceba6595baa445d

It has some of my best animations (I used Blender and all kinds of tricks.) and it's also the source of the image that starts with "An isomorphism is a two-way mapping".