$\newcommand{\E}{\mathcal{E}} \newcommand{\B}{\mathcal{B}}$ I'm currently studying a paper that talks a lot about Grothendieck fibrations and so I'm trying to work with them a bit to get used to them. So I looked at the nLab page to find some properties I should try to prove by myself, and here's one that got me stuck : the nLab page says that if we have a fibration $p:\E\to \B$ and we know how to compute limits in $\B$ and in $\E_A$ (the fiber category over $A$) for all $A\in \B$, then we know how to compute limits in $\E$. This is a very informal statement but hopefully you get what I mean (in the worst case, you can just look up the nLab page)
However I could not fully prove this, so let me show you what I did and where I got stuck. My question is :
How can I go further/ am I on the right track/ did I misunderstand the statement of the nLab page ?
For the sake of simplicity, I'm assuming I have a cloven fibration, that is given $f:A\to B$ in $\B$, I have a functor $f^* : \E_B\to \E_A$ and a family of natural maps $\rho^f_Y : f^*Y\to Y, Y\in \E_B$ such that $\rho^f_Y$ is a cartesian morphism above $f$ whenever $Y$ is above $B$ (of course, naturality is in maps $\E_B$, that is, maps that get sent to $id_B$)
So here's what I did : let $I$ be a category, $F: I\to \E$ a functor. Consider $pF : I\to \B$, assume it has a limit $(L, (\pi_i)_{i\in I})$. Now for $i\in I$ we get a cartesian $\rho^{\pi_i}_{F(i)}: \pi_i^*F(i)\to F(i)$ above $\pi_i$.
Of course $\pi_i^*F(i)\in \E_L$. Moreover, if $f:i\to j$ we get by the fibration property a unique map $\pi_i^*F(i)\to \pi_j^*F(j)$ making the obvious diagram commute : in other words there is a unique functor $\delta : I\to\E_L$ with values $\pi_i^*F(i)$ on objects and such that $i\mapsto \rho^{\pi_i}_{F(i)}$ is a natural transformation $\iota\delta \to F$ (where $\iota$ is the inclusion $\E_L\to \E$).
Now assume we have a cone $(C,(r_i)_{i\in I})$ over $F$. Then $(pC, (p(r_i))_i)$ is a cone over $pF$, so we get a unique $R: pC \to L$ with $r_i = \pi_i\circ R$.
Now fix $i$ and consider the diagram expressing this last equation. By the fibration property we get a unique $R_i : C\to \iota\delta(i)$ such that $\rho^{\pi_i}_{F(i)}\circ R_i = r_i$.
By this uniqueness, we get that $(C,(R_i)_{i\in I})$ is a cone over $\iota\delta$. Notice that I can't say "and now if I have a limit of $\delta$ in $\E_L$ I'm done" because this cone isn't in $\E_L$: $C$ isn't even in $\E_L$ !
And that's where I'm stuck, I can't see how to use the limits in $\E_L$ to get to general limitS; I don't see where the cone in $\E_L$ would come from. What I have is an isomorphism (natural in $C$) $\hom_{\E^I}(\Delta(C), F) \simeq \hom_{\E^I}(\Delta(C), \iota\delta)$ where $\Delta$ is the diagonal functor. Can anyone help ?
(of course it would be much better to draw the diagrams, and I did so at home, but drawing them on here seems really complicated)
