Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it to be "terminal" in some way, or to play a role in factorizing certain (all?) (co)fibrations.
Note: I'm aware that $\mathsf{cod}$ can be easily interpreted via comma categories, but I'm not very familiar with them, so any pointer in that direction would be helpful, too.