CATS Are primes ever generators?

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