There are some ways to introduce groups in Category Theory.

1. Given object $\displaystyle A$ and morphisms $\displaystyle \mu: A \times A \rightarrow A$, $\displaystyle \tau: A \rightarrow A$. We can introduce group on $\displaystyle \mathrm{Hom}(X, A)$ for any object $\displaystyle X$.

Morphisms $\displaystyle \mu$ and $\displaystyle \tau$ have to be such that the respective functions:

$\displaystyle \mathrm{Hom}(X, A) \times \mathrm{Hom}(X, A) \simeq \mathrm{Hom}(X, A \times A) \rightarrow \mathrm{Hom}(X, A)$, $\displaystyle \mathrm{Hom}(X, A) \rightarrow \mathrm{Hom}(X, A)$ satisfy group axioms. The arrows are defined in the natural way.

I think, that 'in the natural way', means that for $\displaystyle f, g \in \mathrm{Hom}(X, A)$ we have $\displaystyle f \cdot g = \mu \circ f \times g$, where $\displaystyle \times$ denotes product of morphisms (which exists due to the definition of product in category). And $\displaystyle f^{-1} = \tau \circ f$.

One problem is, what properties has to satisfy $\displaystyle \mu$ and $\displaystyle \tau$ in order to induce group structure.

But i don't want to think on this right now.

2. The second way, is to suppose we have group structure on each $\displaystyle \mathrm{Hom}(X, A)$ for given $\displaystyle A$ and any $\displaystyle X$. Of course the groups have to be related somehow. And the relation is, that for any morphism $\displaystyle \mathrm{Hom}(Y, X)$ the respective arrow $\displaystyle \mathrm{Hom}(X, A) \rightarrow \mathrm{Hom}(Y, A)$ is a group homomorphism.

Finally the question is. We have group structure given in the way 2, and we want to define group structure in the way 1. It is, we want to define morphisms $\displaystyle \mu$ and $\displaystyle \tau$ in some way.