I have just started learning category theory and I am trying to get an understanding of how to think about the Yoneda lemma. Obvious applications are clear to me (Yoneda embedding is full and faithful, for example), but I want to understand how one uses it in practical mathematics. In particular, I have the following question:
How exactly does Yoneda imply that the definition of a group object in a category with finite products (as an object $G$ together with a functor from that category to the category of groups such that the composite of it with the forgetful functor to the category of sets is represented by $G$) is equivalent to requiring the natural diagrams that give the group axioms to commute?
I am really searching for some intuition. Thanks for any help.