CATS Are primes ever generators?

[David Espinosa <espinosa@kestrel.edu>]

We can say that an object P in a category with coproducts is *prime*
if whenever f : P -> A+B, f factors through one of the injections into
A+B.

(1) I didn't find any reference to this (obvious) notion of
    primality in the standard texts.  Does it occur anywhere?

(2) Is there any condition on the category under which the set of
    primes is a generating family?  Since objects are decomposable
    into a "quotient of a coproduct of generators" (Borceux, volume 1,
    page 151), this would give a decomposition into primes.

Thanks,

David