Collection of subgroups The question revolves around a statement like the following:

Let $G_\lambda \  (\lambda \in \Lambda)$ be subgroups of $G$...

How do I collect subgroups of $G$ in a mathematically precise manner? Do I interpret $G_\lambda \ (\lambda\in \Lambda)$ as a function
\begin{align}
\lambda \mapsto G_\lambda
\end{align}
such that for each $\lambda \in \Lambda$, where $\Lambda$ is some index set, I get a unique subgroup of $G$, denoted by $G_\lambda$. But then if I interpret it as a function, what is its codomain?
Or do I interpret $G_\lambda \ (\lambda \in \Lambda) $ as 
$$
\lambda \mapsto \tau_\lambda \in \text{Mor}(H,G)
$$
where 
$$
\tau_\lambda : H \to G
$$
a monomorphism in the category of groups? (monomorphism, in fact,  for then $\tau_\lambda(G_\lambda)$ is indeed a subgroup of $G$). 
Finally, I would like to be able to understand the following statement:

Let A be any subset (not subgroup) of a group $G$. The intersection of all subgroups of $G$ which contain $A$ itself contains $A$ and is the subgroup of $G$ generated by $A$.


It is usually referred to as a family which is exactly the same thing as a function.
Its codomain is $\{G_\lambda, \lambda\in \Lambda\}$, which is a set according to the axiom scheme of replacement. Alternatively, you could say its codomain is $\mathcal{P}(G)$, the powerset of $G$, or if you want the image $\{H\in \mathcal{P}(G)\mid\exists \lambda, H= G_\lambda\}$.
The last statement has not much to do with the previous question. It just says that for any subset $A\subset G$, there is a least subgroup of $G$ that contains it, usually called the subgroup generated by $A$. There are often different descriptions of this subgroup in terms of $A$.
