The Stacks project

Lemma 4.34.1. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ and $p' : \mathcal{S}' \to \mathcal{C}$ be fibred categories. Let $F : \mathcal{S} \to \mathcal{S}'$ be a $1$-morphism of fibred categories over $\mathcal{C}$. Consider the category $\mathcal{I}_{\mathcal{S}/\mathcal{S}'}$ over $\mathcal{C}$ whose

  1. objects are pairs $(x, \alpha )$ where $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ and $\alpha : x \to x$ is an automorphism with $F(\alpha ) = \text{id}$,

  2. morphisms $(x, \alpha ) \to (y, \beta )$ are given by morphisms $\phi : x \to y$ such that

    \[ \xymatrix{ x\ar[r]_\phi \ar[d]_\alpha & y\ar[d]^{\beta } \\ x\ar[r]^\phi & y \\ } \]

    commutes, and

  3. the functor $\mathcal{I}_{\mathcal{S}/\mathcal{S}'} \to \mathcal{C}$ is given by $(x, \alpha ) \mapsto p(x)$.

Then

  1. there is an equivalence

    \[ \mathcal{I}_{\mathcal{S}/\mathcal{S}'} \longrightarrow \mathcal{S} \times _{\Delta , (\mathcal{S} \times _{\mathcal{S}'} \mathcal{S}), \Delta } \mathcal{S} \]

    in the $(2, 1)$-category of categories over $\mathcal{C}$, and

  2. $\mathcal{I}_{\mathcal{S}/\mathcal{S}'}$ is a fibred category over $\mathcal{C}$.

Proof. Note that (2) follows from (1) by Lemmas 4.33.10 and 4.33.8. Thus it suffices to prove (1). We will use without further mention the construction of the $2$-fibre product from Lemma 4.33.10. In particular an object of $\mathcal{S} \times _{\Delta , (\mathcal{S} \times _{\mathcal{S}'} \mathcal{S}), \Delta } \mathcal{S}$ is a triple $(x, y, (\iota , \kappa ))$ where $x$ and $y$ are objects of $\mathcal{S}$, and $(\iota , \kappa ) : (x, x, \text{id}_{F(x)}) \to (y, y, \text{id}_{F(y)})$ is an isomorphism in $\mathcal{S} \times _{\mathcal{S}'} \mathcal{S}$. This just means that $\iota , \kappa : x \to y$ are isomorphisms and that $F(\iota ) = F(\kappa )$. Consider the functor

\[ I_{\mathcal{S}/\mathcal{S}'} \longrightarrow \mathcal{S} \times _{\Delta , (\mathcal{S} \times _{\mathcal{S}'} \mathcal{S}), \Delta } \mathcal{S} \]

which to an object $(x, \alpha )$ of the left hand side assigns the object $(x, x, (\alpha , \text{id}_ x))$ of the right hand side and to a morphism $\phi $ of the left hand side assigns the morphism $(\phi , \phi )$ of the right hand side. We claim that a quasi-inverse to that morphism is given by the functor

\[ \mathcal{S} \times _{\Delta , (\mathcal{S} \times _{\mathcal{S}'} \mathcal{S}), \Delta } \mathcal{S} \longrightarrow I_{\mathcal{S}/\mathcal{S}'} \]

which to an object $(x, y, (\iota , \kappa ))$ of the left hand side assigns the object $(x, \kappa ^{-1} \circ \iota )$ of the right hand side and to a morphism $(\phi , \phi ') : (x, y, (\iota , \kappa )) \to (z, w, (\lambda , \mu ))$ of the left hand side assigns the morphism $\phi $. Indeed, the endo-functor of $I_{\mathcal{S}/\mathcal{S}'}$ induced by composing the two functors above is the identity on the nose, and the endo-functor induced on $\mathcal{S} \times _{\Delta , (\mathcal{S} \times _{\mathcal{S}'} \mathcal{S}), \Delta } \mathcal{S}$ is isomorphic to the identity via the natural isomorphism

\[ (\text{id}_ x, \kappa ) : (x, x, (\kappa ^{-1} \circ \iota , \text{id}_ x)) \longrightarrow (x, y, (\iota , \kappa )). \]

Some details omitted. $\square$

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