TITLE: Is there a term for a morphism which is surjective on generalized points?
QUESTION [5 upvotes]: Let $C$ be a category. Recall that a morphism $f : a \to b$ in $C$ is said to be a monomorphism if, for any morphisms $g_1, g_2 : c \to a$, it is true that $f g_1 = f g_2$ implies $g_1 = g_2$. Equivalently, $f$ is a monomorphism if and only if it is injective on generalized points in the sense that the induced map $\text{Hom}(c, a) \to \text{Hom}(c, b)$ given by composition with $f$ is an injection for all $c$. 
Is there a corresponding term for morphisms which are surjective on generalized points? Note that any such morphism is a retraction, hence an epimorphism. Indeed, the induced map $\text{Hom}(b, a) \to \text{Hom}(b, b)$ is surjective, so there exists $g \in \text{Hom}(b, a)$ such that $fg = \text{id}_b$. But the converse fails since there exist epimorphisms which are not retractions, such as the quotient $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ in $\text{Ab}$.

REPLY [4 votes]: Oh, ha! These morphisms are precisely the retractions. If there exists $g \in \text{Hom}(b, a)$ such that $fg = \text{id}_b$, then for every $h \in \text{Hom}(c, b)$ it follows that $gh \in \text{Hom}(c, a)$ maps to $h$ under the map induced by $f$. That's curious. 
Retractions are also known as split epimorphisms, so I suppose that's my answer.