Let me start by specifying that I am a total layman with regard to category theory. I was reading this introductory paper and was struck by this sentence in the introduction:
How is the lowest common multiple of two numbers like the direct sum of two vector spaces? What do discrete topological spaces, free groups, and fields of fractions have in common?
It made me wonder if this branch of pure mathematics has allowed us to recognize non trivial isomorphisms between entities/sets that we would not have have recognized otherwise. Can you elucidate a non-expert on what are the most improbable isomorphisms identified by the lenses of category theory?
Surely this sentence is a great advertisement of category theory.
Category theory was invented for studying so called natural isomorphisms. The first paper in the subject was titled General Theory Of Natural Equivalences. Of course in the realm of category theory natural has a precise mathematical meaning, which subjectively may deviate from the usual, common language meaning of the word (but I don't think this is often the case).
I would say that isomorphisms one detects utilizing the machinery of category theory are the most important ones in mathematics. Stefan Banach once said that:
I think that (natural) isomorphisms of category theory are among the most accessible (when they are revealed by some ingenious person) for human cognition. The reason is that human mind have this peculiar ability of grasping the concepts by means of analogies.
As for concrete and interesting examples of an isomorphism that can be identified via categorical methods there is a short and a very conceptual proof of a celebrated theorem of functional analysis, which greatly indicates the importance of categorical methods.
Moreover, existence and rapid development of whole branches (algebraic topology and algebraic geometry are prime examples) of modern mathematics is hard to imagine without linguistic advantages of the categorical methods.