ct.category theory - Equivalence of definitions of cartesian morphisms

Let $p: Cto D$ be a functor, and let $f:yto x$ be a morphism of $C$. We say that $f$ is cartesian if the canonical map $Q:(Cdownarrow f) to P:=(Cdownarrow x)times_{(Ddownarrow p(x)} (Ddownarrow p(f))$ is a surjective (on objects) equivalence of categories. However, if we write out what the (strict 2-) pullback means, the objects are precisely the pairs of morphisms $g: zto x$ and $h:p(z)to p(y)$ such that $p(g)=p(f) circ h$. If we look at the fibres of $Q$ over objects of $A$, we see that that they are contractible groupoids.

Using the more common definition of a cartesian morphism, we must show that any pair of morphisms $(g, h)$ as above uniquely determines an arrow $ell:zto y$ such that $fcirc ell= g$ and $p(ell)=h$.

I see how the first definition implies the existence of such a map, but how does it determine the map's uniqueness (up to more than a contractible space of choices)?