# Difference between initial and terminal objects in a category

that have an outgoing morphisms to every set

(except the empty set)

That's the key part. The requirement is that there be outgoing morphisms to every set **full stop**. The existance of one set where this breaks down is enough to undermine it. That's why, indeed, the initial object in `Set`

is already well-defined without even requiring uniqueness: the empty set is *the only* set that has an outgoing arrow to the empty set.

Meanwhile, every non-empty set has incoming arrows from truely every set including the empty one, but only for one-element sets is this arrow unique.

Here's the direct quote from the book: "The initial object is the object that has one and only one morphism going to any object in the category." Notice the *only one* part.

BTW, I was also careful not to say "any *other* object," because there also is a unique morphism from the initial object to itself: it's always the identity.