I'm reading Kai Behrend's wonderful Introduction to Algebraic Stacks from the London Mathematical Society lecture notes on Moduli Spaces.
I am trying to understand the proof of Lemma 1.75, which says that if a groupoid fibration admits a versal family, then it is a prestack, but I am confused about how the definitions fit together with the proof.
In Behrend's notes, he defines a prestack by first constructing the topological space $Isom(x,y)$. Here $x/S, y/T$ are objects in the groupoid fibration $\mathfrak{X}$/Top, and as a set $Isom(x,y)$ consists of triples $(t, \phi, s)$, where $t\in T$, $s\in S$, and $\phi: x_t \to y_s$ is an isomorphism. (Here $x_t$ is the pullback of $x$ along the inclusion $\{t\}\to T$.) The topology on $Isom(x,y)$ is the finest topology for which, given any space $U$, maps $U \to S, U\to T$, and an isomorphism $\phi: x|_U \to x|_U$ in $\mathfrak{X}(U)$, the corresponding map $U\to Isom(x,y)$ is continuous.
His definition of a prestack is then a groupoid fibration in which, for any two objects $x,y$ in $\mathfrak{X}$, we have:
(i) the natural projections from $Isom(x,y)$ to $T$ and $S$ are continuous
(ii) for any continuous map $\alpha: U \to Isom(x,y)$, there exists a unique isomorphism of families $\phi: x|_U \to y|_U$ giving rise to $\alpha$, using the recipe from the previous paragraph.
(He mentions that this definition is nonstandard, but I’m going to use it in what follows anyway.) I’m confused by the first step in the proof. He says that, in order to prove that $\mathfrak{X}$ is a prestack, it suffices to prove there exists, for each $x,y\in\mathfrak{X}$, a topological space $I$ and a 2-cartesian diagram
(The underlines here turn a topological space $T$ into a groupoid fibration $\underline{T}$ in which the objects are spaces $S$ together with maps $S\to T$ and the morphisms are continuous maps which commute with the given maps to $T$.)
It seems as though we’re supposed to conclude that, if this is true, it must also be true with $Isom(x,y)$ in the place of $I$, but I don’t see why. How do I make the leap from the existence of such an $I$ to the definition of prestack given in the text?