This is a question about the category Grp (groups).
The book "Chapter 0" by Aluffi says that the objects of the category are groups, and the morphisms homomorphisms. He then says that we need not mention that the objects (groups) contain an identity and a unique inverse for every element because all these properties are already contained within the definition of the morphism given: in that $f(1_G)=1_H$ and $f(a^{-1})=(f(a))^{-1}$.
This confuses me. Even if homomorphisms clearly have these properties, do we still not need to mention that a group HAS to contain an identity and inverses for each element? What the above argument suggests is that IF a group has an identity, then it maps to the identity of another group, and so on.
Thank you.
We can take either point of view: Either groups have identities and inverses that are preserved by homomorphisms, or homomorphisms take identities to identities and inverses to inverses (so there must be identities and inverses in every image).
This perhaps is better motivated by the observation that category theory doesn't actually need the objects. A "just-arrow" category is equivalent. If you go the just-arrow route, then the conserved properties of the morphisms have to be baked into the arrows.