What do I need to prove in order to show that a map is an isomorphism?

Question

I need to show that a diffeomorphism from a differentiable manifold $M$ to a differentiable manifold $N$ is an isomorphism. We have defined a diffeomorphism to be a $C^{\infty}$ homeomorphism with a $C^{\infty}$ inverse.

But, as I am new to pure mathematics, I don't know what it means for something to be an isomorphism(structure-preserving map would be my guess, whatever that means precisely) and I don't know what I need to show in order to prove that the map is an isomorphism.

Any help will be appreciated.

Answer 1

In differential topology, isomorphism can be taken to mean homeomorphism, ie. Isomorphism between topological spaces. An isomorphism between topological spaces is a continuous bijection with continuous inverse.