Definition biproduct (category theory) Consider the following fragment found on https://en.wikipedia.org/wiki/Biproduct
For convenience, here is an image of the relevant section:

Can someone explain how this definition implies that there is a zero object? I can't see why the empty set corresponds to a zero object.
It this is a mistake, then what is the 'correct' definition?
 A: The last sentence explains how the empty biproduct is a zero object. Namely the empty biproduct, I'll call it $0$, is an empty product, thus a terminal object, and an empty coproduct, thus an initial object. Since it is both a terminal and initial object, it is a zero object.
I'll note that there is actually a relevant error that might be confusing you, namely that the conditions on the morphisms only imply that the biproduct is both a product and coproduct in the case when the biproduct is not empty. If we leave out this part of the definition in the empty case, then the definition would read "a biproduct is any object together with a collection of no morphisms," which is not what we want.
A: When $n=0$, the given definition reads "A biproduct of the empty family is an object $\oplus_{i\in \emptyset} A_i$ in $C$ together with no morphisms, such that $\oplus_{i\in \emptyset} A_i$ is a product of the empty family and $\oplus_{i\in \emptyset} A_i$ is a coproduct of the empty family." A product of an empty family of objects is readily shown to be a terminal object, since by definition, if $P$ is a product of an empty family then for every $A$, given no morphisms out of $A$ there is a unique morphism $A\to P$ making no triangles commute. Similarly, an empty coproduct is an initial object, so $\oplus_{i\in \emptyset} A_i$ is a zero object.
