It is known that the *codomain fibration* is given by a functor in the form $\mathcal{C}^{\rightarrow}\longrightarrow\mathcal{C}$ where $\mathcal{C}$ is a category having pullbacks and $\mathcal{C}^{\rightarrow}$ is the *arrow category*. It is also known (see B. Jacobs, *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics 141, North Holland, Elsevier, 1999. ISBN 0-444-50170-3) that the condition of $\mathcal{C}$ having pullbacks is necessary and sufficient.

In the same book can also be found the definition of a chain of fibrations in the form $\mathcal{C}^{\rightarrow\rightarrow}\longrightarrow\mathcal{C}^{\rightarrow}\longrightarrow\mathcal{C}$ etc and that it depends again on the condition of $\mathcal{C}$ having pullbacks.

My question is: could this sequence be defined as a fibration when $\mathcal{C}$ is just a cartesian category? Would the fibers be the same (that is: the slices over an object in $\mathcal{C}$)?