I'm following Vistoli's note on fibered categories and stacks. There Proposition 3.11 he states, which as I understand is a classical result with many references, which says $-$
A fibered category over $\mathcal{C}$ with a cleavage defines a pseudo- functor on $\mathcal{C}$.
But as is the issue with such classical results, I'm yet to find a full proof of this fact, and everyone mostly gives sketches, as does Vistoli. I decided to write a whole proof myself, and I'm having some issues :
- consider a fibered category $F:\cal{F}\to\cal{C}$ with a cleavage. This means we have unique pullbacks for every morphism in the base. With this information, I define a pseudofunctor $\Phi : \cal{C}\to Cat$.
- For an object $x\in \cal{C}$, $\Phi(x)$ is just the fiber category of the functor $F$ over the object $x$ and the morphism $id_x$.
- For a morphism $a:x\to y$, $\Phi(a):\Phi(y)\to\Phi(x)$ is a functor. We take any object $B$ over $y$, and using the morphism $x\to y$ we can pullback $B$ to an object $A$ over $x$, using our cleavage. This defines $\Phi(a)$ on objects, and on morphisms, if $\alpha:B_1\to B_2$ is a morphism, then $\Phi(a)(\alpha):\Phi(a)(B_1)\to \Phi(a)(B_2)$ is the morphism which can be defined as a lift; as $\Phi(a)(B_2)\to B_2$ is cartesian.
- Now to the $2$-morphisms. Let $x\xrightarrow{a}y\xrightarrow{b}z$ be two composable arrows. We have to define a natural transformation between functors $\Phi(a)\Phi(b)\implies \Phi(ba):\Phi(z)\to \Phi(x)$. This is the part I'm having issues with. I have been able to define a morphism between $$\Phi(a)\Phi(b)(Z)\to \Phi(ba)(Z)$$ for any object $Z$ in $\Phi(z)$. But I don't know how to show this actually gives a natural transformation. As in now I would have to show the naturality square for this natural transformation commutes. For this I would need to choose a morphism $Z_1\to Z_2$ in $\Phi(z)$ and look at the relevant square. I feel like this again would require the cartesianness of $\Phi(ba)(Z_2)\to Z_2$ but not sure how to write a complete proof. To apply the cartesian property, I'm unable to show that there is a factorization downstairs, which I wanted to lift upstairs uniquely, and show both my morphisms are factorizations, hence they're the same by uniqueness.
Can anyone help me with some hints or references? Thanks in advance.