a property of global elements in the category of sets In the category $\operatorname{Set}$, choose a singleton $*$.  Since $*$ is a terminal object, a morphism $e: * \to X$ is also known as a global element of $X$.  Here the global elements determine the morphisms from $X$, in the sense that they have the following property: two morphisms $f$, $f': X \to Y$ are equal if and only if $fe = f'e$ for all $e: * \to X$.
Does this property have a name?  Can we describe the categories that have it?
 A: An object $X$ with the property that $\text{Hom}(X, -)$ is faithful is called a generator or a separator. Categories such that the terminal object is a generator are somewhat rare (and I don't know a term for this condition); examples include $\text{Top}$ and the category of varieties over an algebraically closed field $k$. Nonexamples include any nontrivial abelian category (since here the terminal object is the zero object) and the category of rings (since here the terminal object is the zero ring).
Even very small perturbations of $\text{Set}$ don't have this property. For example, in the category $G\text{-Set}$ of $G$-sets ($G$ some group), the functor $\text{Hom}(1, -)$ is the functor of $G$-invariants, and this is almost never faithful. Similarly, in the category $\text{Sh}(X)$ of sheaves of sets on a topological space $X$, the functor $\text{Hom}(1, -)$ is the functor of global sections, and this is again almost never faithful. 
Edit: In topos theory a slightly stronger condition is called being well-pointed. 
