Natural numbers object via initial morphism

Question

I assume that a natural number object (or see nLab) can be defined as an initial morphisms.

(edit: as in the title, I ment initial morphism, not objects)

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Thoughts: Probably $X:=1$, and the object map of $U(N)$ is the digram... but no, in this case it wouldn't be "for every '1 over q to A over f to A", becuase if the diagram is encoded in the functor, the universal quantification isn't over the $f$. Also, how would the initial morphism definition enforce that there is only a single $u$ involved in the recursion scheme that's being produced?

I probably only need to know the right functor and then it's done.

Answer 1

An NNO is an initial object in the category of algebras for the endofunctor $X \mapsto 1+X$.

Answer 2

You want to look at the category of diagrams

$$ 1 \to A \to A $$

where the two copies of $A$ should be interpreted as being the same object, rather than two separate objects of the index category that happen to map to the same object.