A typical classification result for a class $C$ of objects looks like that:

**Theorem.** *Each object of $C$ is isomorphic to one object of the following list: [insert list here].*

Examples are the <a href="https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Statement_of_the_classification_theorem">classification of finite simple groups</a> and the <a href="https://en.wikipedia.org/wiki/Surface_(topology)#Classification_of_closed_surfaces">classification theorem of closed surfaces</a>.

Now I noticed sometimes people also call theorems of the following type "classification results". Here, $I$ is a sufficiently strong invariant of the class $C$ of objects.

**Theorem.** *Let $X$ and $Y$ be two objects of $C$. Then $X\cong Y$ if and only if $I(X)= I(Y)$.*

**Question:** In which way are the two different types of "classification theorems" related? Does a classification theorem of the first type imply a classification theorem of the second type or vice versa?