What are the $2$-morphisms in the $2$-category of "categories over $\mathfrak{Sch}$"? Let $\mathfrak S$ be the category of schemes. My goal is just to visualize the $2$-morphisms in the $2$-category consisting of the following objects: categories over $\mathfrak S$, i.e.
$$
\textrm{ categories } X, \textrm{ together with a (covariant) functor } X\to \mathfrak S.
$$ 
(I'm following Fantechi's paper "Stacks for everybody", where at some point I get lost.) 
So, suppose we have two "$0$-objects" (functors) $X\to\mathfrak S$ and $Y\to\mathfrak S$; then we have a category $\textrm{hom}_\mathfrak S(X,Y)$ whose objects are the commutative triangles that one expects (and that I am unable to draw here). My trouble is in understanding the morphisms in this category, i.e. the so called "$2$-morphisms". Such morphisms should of course be natural transformations $\phi\to\psi$ of functors $\phi,\psi:X\to Y$ over $\mathfrak S$. It is exactly this condition "over $\mathfrak S$" that I cannot translate. More precisely, in the paper it is written that a morphism in $\textrm{hom}_\mathfrak S(X,Y)$ is
$$
\textrm{"a natural transformation over the identity functor on } \mathfrak S"
$$
Can anyone explain to me the part "over the identity functor"?
Thanks!

Let $\mathcal{C}$ be a category. The standard definition of 2-morphism in $(\mathfrak{Cat} \downarrow \mathcal{C})$ is this: given objects $P : \mathcal{D} \to \mathcal{C}$ and $Q : \mathcal{E} \to \mathcal{C}$ and functors $F, G : \mathcal{D} \to \mathcal{E}$ such that $Q F = P = Q G$, a 2-morphism $\alpha : F \Rightarrow G$ is a natural transformation $\alpha : F \Rightarrow G$ such that, for all objects $d$ in $\mathcal{D}$, the morphism $Q (\alpha_d) = \textrm{id}_{P d}$. This is what is meant by "a natural transformation over the identity functor". Another name for it is "vertical natural transformation", because the morphisms $\alpha_d$ are "vertical" in the sense of being contained in the fibre of $Q$ over $P d$.
