We define a category $C$ to be $Grp$-Enriched if for every $X$,$Y\in C$, we have that $C(X,Y)$ 'has group structure'/is a group. But what does that mean really? If I am given any finite set $A$, there is a group of order $|A|$, $Z_n$, so may I say "A has a group structure"? But I imagine that's not what they mean when they say the hom sets have group structure. But then 'where' is this structure? Is there a canonical way this structure is imposed across all hom sets? Because if $C(X,Y)$ is just a set of morphisms, and we say we can look at it as a group, isn't that true of every set?
I feel like I'm just looking at it in the wrong way. I suppose a related question (which I think might underscore my confusion better), when we say $G$ is an object in the category $Grp$, and thus $G$ is a group. How are we looking at $G$? Are we looking at $G$ to be a set G with a product map $\bullet$? Or am I stuck in set theoretic ways of thinking?