In category theory there is the notion of a set $\mathcal S$ of generators for  a category $\mathcal C$. The defining property is that  for any morphisms $f,g:A \to B$ in $\mathcal C$ $f=g$ if and only $fa=ga$ for all morphisms $a:S \to A$ for all $S \in \mathcal S$. 

Thus for a set of generators morphisms with the same domain and codomain in $\mathcal C$ can be distinguished by morphisms from elements of $\mathcal S$. 

A more subtle property which is especially relevant to algebraic categories is that of _density_. A small subcategory $D$ of $\mathcal C$ is dense in $\mathcal C$ of for any objects $A,B$ of $\mathcal C$ the natural function 

$$\mathcal C(A,B) \to Nat_D(\mathcal C(-,A), \mathcal C(-,B))$$

which assigns to a morphism $f$ the natural transformation of functors $D\to \mathcal C$ which sends $\mathcal C(-,A) \to \mathcal C(-,B)$ by composition. Intuitively, this says that morphisms $A \to B$ can be recovered from morphisms from objects of $D$. 

A   dense subcategory  of the category of groups is that generated by the free groups on a 1 and 2 elements. 

For more discussion, see an article by Vaughan Pratt http://boole.stanford.edu/pub/yon.pdf.