First a note about terminology: older literature sometimes uses the term "cofibration" for a functor $p:E\to B$ such that $p^{\rm op} : E^{\rm op} \to B^{\rm op}$ is a fibration, but it's more common nowadays to call this an "opfibration", with "cofibration" used instead for a functor whose corresponding morphism $B\to E$ in ${\rm Cat}^{\rm op}$ is a fibration. One reason for this is that opfibrations, like fibrations, are characterized by a lifting property, like fibrations in homotopy theory, whereas cofibrations (in the modern sense) are instead characterized by an *extension* property, like cofibrations in homotopy theory.

With that out of the way, the most basic universal property of $\rm cod$ is that it is the free opfibration over $C$ generated by ${\rm id} : C\to C$. In other words, if $p:E\to C$ is an opfibration equipped with a section $s:C\to E$, such that $p s = 1_C$, then there is an essentially unique map of opfibrations $\tilde{s} : C^{\to} \to E$. More generally, for any functor $f:A\to C$, the opfibration $A\times_C C^{\to}$ is the free opfibration generated by $f$. This doesn't even need the fact that $C$ has pullbacks.

A fancier universal property, which does use finite limits in $C$ and the consequent fact that $\rm cod$ is also a fibration, is that in the "universe" of fibrations over $C$, the codomain fibration is the "free coproduct completion of a point", in the same way that $\rm Set$ is the free coproduct completion of a point in the universe of ordinary categories. Here "coproduct complete" means having "indexed coproducts", meaning that a fibration is also an opfibration and satisfies the Beck-Chevalley condition. Thus, for any fibration $p:E\to C$ with this property, and any object $X\in p^{-1}(1)$, there is an essentially unique $g:C^{\to}\to E$ that is a morphism of both fibrations and opfibrations and maps $1_1$ to $X$.