As the title suggests, I am really confused about the canonical isomorphisms that show up in the diagram in the following construction.
Let $C$ be a category with finite fiber products, and $p: F \to C$ a fibered category. Choose for each $f: X \to Y$ in $C$, a pullback functor $f^{*}: F(Y) \to F(X)$.
For each morphism $f:X \to Y$ define a category $F(f: X \to Y)$ with objects pairs $(E, \sigma)$ with $E \in F(X)$, and $\sigma: pr^*_1E \to pr^*_2E$ an isomorphism in $F(X \times_Y X)$ such that the following diagram commutes:
$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} pr_{12}^*pr_1^*E & \ra{pr_{12}^*\sigma} & pr_{12}^*pr_2^*E & \ra{=} & pr_{23}^*pr_1^*E\\ \da{=} & & & & \da{pr_{23}^*\sigma} \\ pr_{13}^*pr_1^*E & \ra{pr_{13}^*\sigma} & pr_{13}^*pr_2^*E & \ra{=} & pr_{23}^*pr_2^*E. \\ \end{array} $$
Where the morphisms denoted by '$=$' are the canonical isomorphisms, and $pr_{ab}: X \times_Y X \times_Y X \to X \times_Y X$ are the projection morphisms.
I'm confused about what these canonical isomorphisms are, and why for example $pr_{12}^*pr_1^*E$ is canonically isomorphic to $pr_{13}^*pr_1^*E$, but not to $pr_{12}^*pr_2^*E$. Any help would be greatly appreciated.
You have a commuting square $pr_1pr_{12} = pr_1pr_{13},$ and so the two pullbacks are identified. Clearly we cannot replace one of the $pr_1$'s with $pr_2$ in the above.