In some categories, there is an object $X$ such that for any $f, g : Y \to Z$, $f = g$ if $f \circ h = g \circ h$ for all $h : X \to Y$.
- In the category of sets, any terminal object works
- In the category of groups, $\mathbb{Z}$ works (since for any $y \in Y$, we can make a homomorphism $h : \mathbb{Z} \to Y$ by letting $h(1) = y$)
- In the category of graphs with no isolated vertices, the graph $\cdot \to \cdot$ works (since we have a homomorphism for each edge of $Y$)
What is this kind of object called?
This is called a generator of the category. The dual notion is that of cogenerator.
You can also have a set of objects that all together act as you suggest, and then they're called a set of generators (of course if you have a category with products with surjective projections and a set of generators, then you also have a generator - as Arnaud D. pointed out, what you really need is a cone with epimorphic projections)