Group theoretic meaning of natural isomorphisms between certain functors

So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to $G$ specified by $\forall g$ $\in G$ $\exists$ $\phi(1)=g$. Each of these homomorphisms is in 1-1 relation with the elements of $G$.

Consider groups for a moment in the context of category theory, viewing the groups as one object categories and thereof group homomorphism as functors. What would a natural isomorphism mean in group theory terms for G?

My thoughts are that the natural transformations equate to maps between homomorphisms and since the homomorphisms are in 1-1 relation with the elements of G, then a natural transformation is a map between group elements, i.e another element of the group. I'm not entirely convinced by this.

Anyway continuing my reasoning a natural isomorphism is a map between the homomorphisms that has an inverse, the thing is in relation to the elements of G this is every element? Furthermore I'm looking to find an equivalence class defined by this natural isomorphism, is this just a collection of the elements that are each others inverses?

Thanks in advance for any assistance.

Generally speaking, a natural transformation $\alpha:F\Rightarrow G$ (where $F,G:\mathcal{C} \to \mathcal{D}$ are functors) is a collection of arrows $(\alpha_C:FC\to GC)$ of $\mathcal{D}$ indexed by the objects of $\mathcal{C}$ satisfying the condition that $\alpha_{C'}\circ F(c)=G(c)\circ \alpha_C$ for all $c:C\to C'$ in $\mathcal{C}$.

In your context, you only have one object, so a natural transformation between two homomorphisms $\alpha:\varphi_1 \Rightarrow \varphi_2$ is simply an arrow in the category formed by $G$, thus an element $a\in G$; the naturality condition is then simply$$a\varphi_1(n)=\varphi_2(n) a$$for all $n\in \mathbb{Z}$. Now if you take the corresponding elements of $G$, $\varphi_j(1)=g_j$ for $j=1,2$, you get that $\phi_j(n)=g_j^n$, and then the naturality can be rewritten $$g^n_2=ag_1^na^{-1}.$$

This has to hold for all $n\in \mathbb{Z}$, but it is sufficient that it holds for $n=1$.

So two functors being naturally isomorphic simply means that the corresponding elements are conjugated in $G$, and a natural isomorphism is any element that realizes this conjugation. Equivalent classes of isomorphic functors correspond to conjugacy classes in $G$.

Let $G$ be a group, $a,b\in G$ be its elements, $f_a,f_b\colon\mathbb{Z}\to G$ be corresponding homomorphisms. Then for every $c\in G$, such that $b=cac^{-1}$, the assigning $*\mapsto c$ is a natural transformation $\alpha_c\colon f_a\to f_b$. Note, that every such natural transformation is a natural isomorphism, because every group is obviously a groupoid, i.e. every its morphism is an isomorphism. Hence the category $G^{\mathbb{Z}}$ of functors from $\mathbb{Z}$ to $G$ is a groupoid and its isomorphism classes are conjugacy classes of the group $G$. For example, if the group $G$ is abelian, then $$ G^{\mathbb{Z}}\cong\bigsqcup_{g\in G}G, $$ where the coproduct is taken in the category of groupoids.