I'm fairly new to category theory (so forgive me if I misstate a fact), but I have noticed the trend that, quite often a single construction can be re-expressed in "seemingly" different ways. For example, if you have a universal property, then it's likely the case that you have an adjunction. But lurking behind every adjunction is a representable functor somewhere. So, you have your pick! You're free to choose whichever language - universal properties, adjunctions, representability - you prefer. (Not a representability person? No problem! Here's an adjunction for you....) And of course, there's the joke that "all theorems are Yoneda."
Now I put "seemingly" above in quotation marks since, as Saunders Mac Lane noted, "The notion of Kan extensions subsumes all the other fundamental concepts of category theory." And indeed, we learn that
- (co)limits (and their universal properties) are really Kan extensions
- every adjunction is really a Kan extension
- the colimit formula of a pointwise Kan extension involves the category of elements $\int F$ of the functor, say $F$, that you're extending. But $\int F$ is precisely the category that informs us of whether or not $F$ is representable*
- Yoneda's Lemma is a consequence of the fact that every functor is its own right Kan extension along the identity
In light of this, I'm prompted to ask: What is the significance of having a formalism in which there are a myriad of equivalent ways of saying the same thing?
In other words, what does being able to describe the same ideas in different, yet equivalent, ways tell you about your theory? Does it simply mean you've done a good job? (So, pat yourself on the back!) Or does it mean you're being redundant and should do away with some of your definitions? Or does it mean that something inherently deep/significant is going on?
Any thoughts would be appreciated!
$^*$ $F$ is representable precisely when $\int F$ has an initial (or terminal, depending on the variance of $F$) object.
Being able to describe the same object in different ways is one of the key ideas in maths. The reason for that is that each different characterization can be useful for different things.
For instance it is usually easier to prove that a given functor has an adjoint using the universal property.
On the other hand using the characterization of adjoints as representable functors allows to use yoneda stuff to prove properties about adjoints in an easy way: for instance it is incredibly easy to prove that adjoints to a given functor are naturally isomorphic using the language of representable functors.
Adjoint functors via zig-zag identities are the key to understand the concept of adjoints 1-cell in every 2-category and it is also useful for working with monads.
These are just few examples to show how much it is important having different characterizations of the same concept, I hope they give you the idea.