In most areas of math, an interesting topic is to (try to) classify in some sense the structures that are studied in such area. For example, in topology people might want to study what kind of topological spaces are there up to homeomorphism. Often the objects of study naturally belong to some (n-)category.
In linear algebra, morphisms between finite dimensional vector spaces also have a classification of sorts. One is up to equivalence. This one actually arises from considering the category whose objects are linear maps and the morphisms are commutative squares. This might suggest that it'd be natural to think that the behaviour of a linear map is "essentially the same" as any other equivalent to it.
However, when the domain is the same as the codomain, there is also the notion of similarity of linear maps. And, worringly, this notion is strictly stronger than equivalence! This implies that thinking of equivalent linear maps as "essentially the same" is wrong, because equivalent linear maps might not be similar.
Is there any natural way to define what it'd mean to classify morphisms of some $1$-category? (in particular, linear maps of finite dimentional vector spaces) Is there a way to conceptually think of morphisms in the same way as we think of objects?