Reading about group objects in categories, it's a fact that a group object is in the category of $\mathcal{Set}$ just a common group. I am trying to give an actual proof of this, but I'm a bit confused on how to connect the group axioms to the definition of a group object - mainly because the group axioms work with elements inside of a group, but in category theory, the elements of an object don't really play a role. So I wonder how to actually proof that a group object in $\mathcal{Set}$ is indeed a group.
The definition of a group object as I know is the following, given by Richard Pink in "Finite group schemes":
A (commutative) group object in the category $\mathcal{C}$ is a pair consisting of an object $G \in ob(\mathcal{C})$ and a morphism $\mu : G \times G \to G$ such that for any object $Z \in ob(\mathcal{C})$ the map $G(Z) \times G(Z) \to G(Z)$, $(g, g') \mapsto \mu \circ (g, g')$ defines a (commutative) group.
Where $G(Z)$ is the set of morphisms $Z \to G$.
Im also not sure if this is a proper definition (e.g. shouldn't it be explained what is ment by "define a group").
Take a group $G$. Then, pointwise multiplication certainly embeds $\hom_{\mathsf{Set}}(Z,G)$ with a structure of group for any set $Z$.
Conversely, if $G$ is group object of set, you can take advantage of the fact that $$ G \simeq \hom_{\mathsf{Set}}(1,G) $$ (where $1$ is a final object of the category $\mathsf{Set}$, that is a singleton).