# Is projection epimorphism?

Let $$\mathscr{C}$$ be category with finite product, and $$A,B$$ be objects of $$\mathscr{C}$$.

And $$p: A\times B \to A$$ is projection.

Now, is $$p$$ epimorphism?

I think it is true, but I canâ€™t proof. Help me, thanks.

Not necessarily. For example take $$C = \text{Set}$$ and take $$B$$ to be empty (and $$A$$ to be non-empty).
For a more interesting example, take $$C = \text{Aff} = \text{CRing}^{op}$$ to be the category of affine schemes and take $$A = \text{Spec } \mathbb{F}_2, B = \text{Spec } \mathbb{F}_3$$; then the fiber product $$A \times B \cong \text{Spec } (\mathbb{F}_2 \otimes \mathbb{F}_3) \cong \text{Spec } 0$$ is empty even though $$A$$ and $$B$$ themselves aren't, and $$p : \emptyset \to \text{Spec } \mathbb{F}_2$$ is not an epimorphism because the dual map on commutative rings $$\mathbb{F}_2 \to 0$$ is not a monomorphism (monomorphisms in commutative rings are precisely the injective maps).
What is true is that if $$B$$ has a global point (a map $$1 \to B$$ where $$1$$ is the terminal object) then $$p$$ has a section and so is a split epimorphism. In particular it's an epimorphism in $$\text{Set}$$ for any non-empty $$B$$, and it's always an epimorphism in any category with a zero object (an object which is both initial and terminal), for example the category of groups.