Is there any way to universally define the notion of $\text{Isomorphism}$? Suppose we want to give a very general definition of the term Isomorphism, first of all, we'll want an isomorphism to be a bijective function.
Informally, we want our function to preserve whatever 'structure' we're talking about, may it be: multiplication, addition, order if we're about ordered fields, or maybe adjacency if talking about graphs, etc.
However, after thinking about this for a few hours, I could not come up with a precise definition of the term which could fit every context.
Do you think such a definition is possible?

I think category theory answers the question. An isomorphism in a category is just a morphism which has a two-sided inverse. 
Now, what is a morphism? This depends on the category you're working on. For example we have:


*

*the category of topological spaces, where continuous functions are the morphisms;

*the category of vector spaces, where continuous linear maps are the morphisms;

*the category of groups, where group homomorphisms are the morphisms;

*the category of differentiable manifolds, where smooth functions are the morphisms;

*the category of sets, where functions are morphisms,

*etc.


For example, homeomorphisms are the isomorphisms in the category of topological spaces (we even see once in a while people say that homeomorphic topological spaces are topologically isomorphic).
