Does it make sense to speak of reindexing functors for fibrations without a cleaving?
A cleaving gives us concrete reindexing functors, but even without a specific cleaving we know that Cartesian arrows exist over each arrow in the base category, so we should also know that reindexing functors exist even without a specific one to point at right?
This came up when digesting the definition of fiberwise structure, since we require the structure to exist in fibres and be preserved by reindexing functors. Can we only speak of fiberwise structure for cloven fibrations?