TITLE: Is the map into the terminal object an epimorphism?
QUESTION [13 upvotes]: Let $C$ be a category with a terminal object $1$. Is the unique arrow from an object into $1$ necessarily an epimorphism? If not, is it an epimorphism if $C$ is a topos?

REPLY [8 votes]: By duality, your question is equivalent to: Is every morphism from the initial object a monomorphism? No, for example in $\mathsf{CRing}$ with initial object $\mathbb{Z}$ this holds only for rings of characteristic $0$.