My Algebra textbook "Chapter 0" by Aluffi states that the category of a group consists of groups as objects and homomorphisms between them as morphisms.
Then it also gives a commutative diagram to encapsulate the properties of a homomorphism: namely that if $f:G\to H$ is a homomorphism, then $f(a*_G b)=f(a)*_H f(b)$.
Is the commutative diagram part of the category theory definition? Or is it an additional property which along with the category theory definition (taking groups as objects and homomorphisms as morphisms) gives the full definition of a group?
Motivation: I thought categories could only contain objects and morphisms. Hence now it seems rather surprising that you could also include special relationships between objects of a category to encpasulate the properties of an Algebraic structure.
Thank you.
This is not a category theory definition of a group, but rather the definition of the category of groups.
Thus, one has to specify objects and morphisms. Objects here are groups, morphisms are homomorphisms of groups defined in the usual sense.
That homomorphisms of groups may be described via maps which make a diagram commute, is completely irrelevant to the definition of the category of groups. However, it offers a generalization of groups, see the notion of a group object.