In category theory I have seen two universal properties for natural numbers. In the category of Peano structures, that is structures such that the objects are triplets $(S,e,f)$ where $S$ is a set, $e$ is a basepoint element and $f$ is an endofunction. A morphism between $(S,e,f)$ and $(T,g,h)$ is a function $\varphi:S\to T$ such that $\varphi(e)=g$ and $\varphi\circ f = h\circ \varphi$. Then in this category the set $\Bbb N$ is the initial object.
In this category the universal property it satisfies basically garantuees, from my understanding at least, that the set has the property of induction to it. That is if $0\in A\subseteq \Bbb N$ and $n\in A\implies S(n)\in A$ then $A=\Bbb N$. I have read a few proofs that shows that the property of induction over natural numbers and the category theory property of being initial in such a category are equivalent.
Now in the category of Unital semirings with unit preserving homomorphisms we have that natural numbers are again an initial object. Which makes sense as how the homomorphisms would work and how multiplication and addition is defined for natural numbers.
Now the Peano category definition gives us the induction property, the Unital Semiring category one gives us the algebraic structure of natural numbers. Neither gives directly the other so my question is more on how we define it categoricly such that we get both?
My thoughts on the matter is that one could use a functor of some sort, seems the most natural doesn't it? I had an idea for a functor from the category of unital semirings etc to the peano category by having for an object $S$ in the semiring category it is sent to the object $(S,0_S,f)$ where $f(x)=x+1$ in the semiring. This works nicely with natural numbers and for any semiring we have this working, as it is unital semirings we are garantueed they have a $1$, naturally I also assume they have a $0$. For a semiring homomorphism $\varphi$ we have that we can take the corresponding set function, as we have $$\varphi(f(x))=\varphi(x+1)=\varphi(x)+1=h(\varphi(x))$$ so it sends homomorphisms of our initial category to morphisms of the other. The case of preserve fixed points is trivial from definition of unital homomorphisms.
I feel intuitively this would distinguish the natural numbers uniquely to include both the algebraic structure and induction property in a more category theoretical sense. My gut feeling is however that there should be a "backward" functor to go from peano to unital semirings. The issue there was I could not define it in any meaningful sense as the successor function could be defined as $f(x)=e$, the constant function to the fixed point, and as such a meaningful addition and multiplication from it would not be possible.
Is my thoughts on this correct or am I off? I know my understanding of adjoint functors is sorely lacking and trying to grasp them, as said my gut feeling it should be here too I just do not know how to do it. Anyhow is this correct? If not are there any good sources for linking the two together?