Regarding what proponents of category theory argue as the purpose of category theory, [nlab] has some extremist takes, [chan] has a video series that shows us how a math student is typically sold on learning category theory, and [bauer] is a video which explores using a corner of category theory to formalize some basic computability results in a very different way to the usual formalism centered on Turing machines.

[nlab/0]: https://ncatlab.org/nlab/show/nPOV

[nlab/1]: https://ncatlab.org/nlab/show/applications+of+%28higher%29+c...

[chan]: https://www.youtube.com/playlist?list=PLgAugiET8rrLkzTFuwmiV...

[bauer]: https://vimeo.com/510188470