TITLE: Categories whose coproducts are not disjoint
QUESTION [5 upvotes]: I’m looking for examples of categories whose coproducts are not automatically disjoint.
A coproduct $X\rightarrow X\coprod Y\leftarrow Y$ is disjoint iff the pullback of the above diagram is an initial object.
I’ve considered Set, Group and Top, but all seem to have naturally disjoint coproducts.
In Group I expected non-disjoint coproducts since the underlying set of the coproduct of groups isn’t a disjoint union, but the pullbacks are singletons just like the initial group.
Any examples are appreciated.

REPLY [11 votes]: Here's an example which is even a familiar category of algebraic objects: in commutative rings the coproduct is the tensor product $\otimes$, and it sometimes happens that the tensor product of two rings is the zero ring. In that case the pullback of the diagram you're looking at is just the product. So, for example, $\mathbb{F}_2$ and $\mathbb{F}_3$ don't have a disjoint coproduct.
(However, coproducts are disjoint in the opposite category of affine schemes!)