In topology, any continuous function can be considered as a *fibration* over space , with total space , where the fibres are glued together according to the topology of .

Likewise, any functor can be regarded as a kind of fibration over . Each object determines a fibre category with arrows which are mapped to .

In this generality, the fibres connect up by means of *profunctors* between them and then and the original composition of are encapsulated in the arising mapping .

More specifically, given any arrow , its preimage determines [is] a profunctor between categories and , all within .

For a composable pair of arrows , we obtain a *comparison map* , which is just the composition defined in of the occuring pairs of arrows.

The comparison maps make this mapping a ‘normal lax functor’.

Under certain hypothesis on , all the arising profunctors become functorial (when each object from the source category has a reflection on the target category), yielding a normal lax functor . Such an is called an opposite *Grothendieck prefibration*. The reflection arrows are also called *(weak) cartesian morphisms* of .

If they are closed under composition in , the comparison maps become invertible, so that composition of the profunctors is preserved (up to isomorphism), then we talk about *Grothendieck opfibration*.

Its dual notion, when all profunctors are opfunctorial, is the *Grothendieck (pre-)fibration*.

Finally, as a bonus, we illustrate here the post Co/lax functors without apriori comparison cells for the general case of an arbitrary functor , by constructing a double profunctor that corresponds to the arising lax functor .

Consider the full sub-bicategory of on certain particular subcategories of as objects, namely, the fibres for each object . Fix a unique vertical heteromorphism for each . All these are only required to support the heterocells which are defined to be the profunctor morphisms , where is an arbitrary profunctor.

The horizontal composition of given heterocells and with arrows of is given by

where , are profunctors and the final composition is performed in .