A biproduct in a preadditive category is an object that combines a category
product and a coproduct with compatible structure
morphisms. For two objects 
and 
, a biproduct 
has injections 
and 
and projections 
and 
. They satisfy 
, 
, 
, 
, and 
. These identities ensure both universal
properties (Stacks Project Authors 2026).
Every additive category has finite biproducts. The empty biproduct is a zero object, an object that is both initial and terminal.
