It should be noted for non-mathematical readers that a group has a single operation called multiplication, and the multiplicative identity is called 1.
The natural numbers form a group with "multiplication" being addition, and "1" being 0. In which case the Grothendieck construction on the natural numbers gives the integers with addition.
If calling the basic operation multiplication instead of addition is confusing, remember that one of the inspirations for group theory are permutations of a set of things, which can be represented by matrices using matrix multiplication to perform the permutations.
The natural numbers form a group with "multiplication" being addition, and "1" being 0. In which case the Grothendieck construction on the natural numbers gives the integers with addition.
If calling the basic operation multiplication instead of addition is confusing, remember that one of the inspirations for group theory are permutations of a set of things, which can be represented by matrices using matrix multiplication to perform the permutations.