I'm starting a course at category theory, and I'm learning about particular categories.
I'm now reading about the $\mathcal{Pno}$ category, where it's objects are the "structures":
$(A,\alpha,a)$, where $A$ is a set, $\alpha: A\to A$, and $a\in A$.
And a $\mathcal{Pno}$-morphism is a function between two structures $(A,\alpha,a),(B,\beta,b)$, $f:A\to B$ such that $$f\circ\alpha=\beta\circ f$$ and $f(a)=b$.
I need to prove that $(\Bbb{N},\mathcal{succ},0)$ is a $\mathcal{Pno}$-object, where $\mathcal{succ}$ is the succesor function.
My question is: Is it just as simple as noting that $\Bbb{N}$ is a set, by definition $\mathcal{succ}:\Bbb{N}\to\Bbb{N}$, and $0\in\Bbb{N}$?