Functor as a fibration

In topology, any continuous function $f:A\to B$ can be considered as a fibration over space $B$ , with total space $A$ , where the fibres $f^{-1}(b)$ are glued together according to the topology of $A$ .

Likewise, any functor $F:\ct A\to \ct B$ can be regarded as a kind of fibration over $\ct B$ . Each object $b\in Ob\ct B$ determines a fibre category $F^{-1}(b)$ with arrows which are mapped to $1_b$ .

In this generality, the fibres connect up by means of profunctors between them and then $F$ and the original composition of $\ct A$ are encapsulated in the arising mapping $\ct B\to \ct Prof$ .

More specifically, given any arrow $\beta:b_0\to b_1\in\ct B$ , its preimage $F^{-1}(\beta)$ determines [is] a profunctor between categories $F^{-1}(b_0)$ and $F^{-1}(b_1)$ , all within $\ct A$ .
For a composable pair of arrows $\beta_1:b_0\to b_1,\ \beta_2:b_1\to b_2$ , we obtain a comparison map $F^{-1}(\beta_1)F^{-1}(\beta_2)\to F^{-1}(\beta_1\beta_2)$ , which is just the composition defined in $\ct A$ of the occuring pairs of arrows.
The comparison maps make this mapping $\ct B\to \ct Prof$ a ‘normal lax functor’.

Under certain hypothesis on $F$ , 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 $\ct B\to\ct Cat$ . Such an $F$ is called an opposite Grothendieck prefibration. The reflection arrows are also called (weak) cartesian morphisms of $F$ .
If they are closed under composition in $\ct A$ , 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 $F:\ct A\to\ct B$ , by constructing a double profunctor $\ct Prof\ag\ct B$ that corresponds to the arising lax functor $\ct B\to\ct Prof$ .
Consider the full sub-bicategory of $\ct Prof$ on certain particular subcategories of $\ct A$ as objects, namely, the fibres $F^{-1}(b)$ for each object $b\in\, Ob\ct B$ . Fix a unique vertical heteromorphism $t_b:F^{-1}(b) \downarrow b$ for each $b\in\, Ob\ct B$ . All these are only required to support the heterocells $\nzt T{t_b}{}{t_{b'}}{\beta}$ which are defined to be the profunctor morphisms $T\to F^{-1}(\beta)$ , where $T:F^{-1}(b)\ag F^{-1}(b')$ is an arbitrary profunctor.
The horizontal composition of given heterocells $\teta:T \to F^{-1}(\beta)$ and $\ro:T' \to F^{-1}(\beta')$ with arrows $b\fl\beta\to b' \fl{\beta'}\to b''$ of $\ct B$ is given by

$TT'\to F^{-1}(\beta\beta');\ \quad\ (t,t_1)\ \mapsto\ t^\teta\cdot t_1^\ro$

where $T:F^{-1}(b)\ag F^{-1}(b')$ , $\,\,T':F^{-1}(b')\ag F^{-1}(b'')$ are profunctors and the final composition $\,\,t^\teta\cdot t_1^\ro\,\,$ is performed in $\ct A$ .