A Morphism that is bijective.

To prove two things are isomorphic, it's enough to find a mapping which is an isomorphism. However, to prove two things are not isomorphic you have to prove that no mapping will be a isomorphism! Because there may be too many maps to check, one most often has to find a relevant Invariant (a property invariant under a certain class of maps), that is different in the two spaces, to prove this.

Some isomorphisms are natural, and others not. Example

An isomorphism in a Category C is an arrow $i:A \to B$ such that there exists an arrow $j: B \to A$ such that $j=i^{-1}$ i.e. $i \circ j = id_A$ and $j \circ i = id_B$.

Gives Bijection for category of sets