When reading the definition of a well-pointed category on for example Wikipedia, only the following condition is given:
- The terminal object 1 is a generator.
However the general definition of a well-pointed topos also mentions:
- The terminal object is not a zero object (i.e. it is not initial at the same time)
I can see how being initial would ruin the odds of being a generator as there is only one unique morphism from the initial object to any other object. But if this was the only reason for including condition 2, then condition 1 would suffice.
Is there a more genuine reason for including condition 2?