If there is an arrow $h: (A\times B) \to (A\times B)$, then, necessarily $h = \left\lt f,g\right\gt$ for some $f: (A\times B) \to A$ and $g (A\times B) \to B$?

The book I'm studying defines a product $A\times B$ to be an object provided with two projecting arrows $\pi_A : A\times B \to A$ and $\pi_B: A\times B \to B$ such that, for any other object C and arrows $f: C \to A$ and $g : C \to B$ there's exactly one arrow $\left\lt f,g\right\gt: C \to (A\times B)$ such that:

$\pi_A \cdot \left\lt f,g\right\gt = f$

$\pi_B \cdot \left\lt f,g\right\gt = g$

It says nothing about $\left\lt f,g\right\gt$ being the only arrow from $C$ to $A\times B$, but I got that impression later when he asked to prove that all products are isomorphic: I could only prove assuming that any arrow from $A \times B$ to itself was of the form $\left\lt f,g\right\gt$ for some f and g from $A\times B$ to A and B, respectively.

This suggests me that $\left\lt f,g\right\gt$ was meant to be the only possible arrow from C to $A \times B$.

Is this right or did I just misunderstood the book?