This question is based on the exercise 1.2.1 from Harold Simmons book, An Introduction to Category Theory.
Consider the category $\textbf{Pno}$ whose objects are $(A,\alpha,a)$, where $A$ is a set, $\alpha:A\rightarrow A$ is a function and $a\in A$ is a distinguished element. The arrows of $\mathbf{Pno}$ are $f:(A,\alpha,a)\rightarrow(B,\beta,b)$, where $f\circ\alpha = \beta\circ f$ and $f(a)=b$.
They asked to prove that there is a unique arrow $f:(\mathbb{N},succ,0)\rightarrow (A,\alpha,a)$, for every object $(A,\alpha,a)$ of $\mathbf{Pno}$, where $succ$ is the successor function.
This arrow is given by $f(n)=\alpha^n(a)$, which verify the properties to be a $\mathbf{Pno}$-arrow. Now, in the solution section they say that a proof by induction proves that this is the only arrow. However, I don't see how an induction (I guess this induction must be done over $n$) proves the uniqueness of this arrow.
Could anybody give me a hand with this? I'd really appreciate it.
Best Answer
Let $g : (\mathbb N, s,0) \to (A,\alpha, a)$ be an arrow. $g(0) = a = \alpha^0(a)$ holds by assumption. Suppose $g(n) = \alpha ^n(a)$ for some $n\in\mathbb N$. Then $g(n+1) = gs(n) = \alpha g(n) = \alpha (\alpha ^n(a)) = \alpha ^{n+1}(a)$. So we have an initial object in the category.