Let $\mathcal{C}$ be any category and $\mathcal{D}$ a fixed category. The trivial fibered category is $\mathcal{F} = \mathcal{C} \times \mathcal{D}$ with $p = \operatorname{pr}_1 : \mathcal{C} \times \mathcal{D} \to \mathcal{C}$. The fiber over any $U \in \mathcal{C}$ is $\mathcal{F}_U \cong \mathcal{D}$. This is the fibered category analogue of a trivial bundle.

Let $\mathcal{C}$ be a category. The arrow category $\operatorname{Arr}(\mathcal{C})$ has morphisms $f : X \to Y$ in $\mathcal{C}$ as objects, and commutative squares as morphisms. The functor $\operatorname{target} : \operatorname{Arr}(\mathcal{C}) \to \mathcal{C}$ sending $f : X \to Y$ to $Y$ gives $\operatorname{Arr}(\mathcal{C})$ the structure of a category over $\mathcal{C}$. The fiber over $Y$ is the slice category $\mathcal{C}/Y$.

Let $\mathcal{C} = \mathbf{Sch}$ (schemes) and let $\mathcal{F}$ be the category of quasi-coherent sheaves: objects are pairs $(S, \mathcal{E})$ where $S$ is a scheme and $\mathcal{E}$ is a quasi-coherent $\mathcal{O}_S$-module. The functor $p(S, \mathcal{E}) = S$.

A morphism $(f, \varphi) : (T, \mathcal{G}) \to (S, \mathcal{E})$ consists of $f : T \to S$ and $\varphi : f^*\mathcal{E} \to \mathcal{G}$. This morphism is cartesian if and only if $\varphi$ is an isomorphism, i.e., $\mathcal{G} \cong f^*\mathcal{E}$. Cartesian arrows correspond to pulling back sheaves.

Let $p : \mathcal{F} \to \mathcal{C}$ be a category over $\mathcal{C}$ where each fiber $\mathcal{F}_U$ is a discrete category (a set). Then every morphism in $\mathcal{F}$ is automatically cartesian. This recovers the theory of presheaves of sets.

Define $\mathcal{F}$ as follows: objects are pairs $(A, M)$ where $A$ is a commutative ring and $M$ is an $A$-module. A morphism $(A, M) \to (B, N)$ is a ring homomorphism $f : A \to B$ together with an $A$-module map $M \to f^*N$ (where $f^*N$ is $N$ viewed as an $A$-module via $f$). The functor $p(A, M) = A$ makes $\mathcal{F}$ fibered over $\mathbf{CRing}$.

The fiber $\mathcal{F}_A$ is the category $A\text{-}\mathbf{Mod}$ of $A$-modules. Given $f : A \to B$ and a $B$-module $N$, the cartesian lift is $(A, f^*N) \to (B, N)$ where $f^*N$ is $N$ with the $A$-module structure $a \cdot n = f(a)n$. This is the restriction of scalars.

Fix a scheme $S$. The category $\mathbf{Sch}/S$ of $S$-schemes is fibered over $\mathbf{Sch}$ via the forgetful functor. More precisely, let $\mathcal{F}$ have objects $(T, X \to T)$ where $T$ is a scheme and $X \to T$ is a morphism of schemes, and let $p(T, X \to T) = T$.

Given a morphism $f : T' \to T$ and an object $X \to T$, the cartesian lift is the base change $X \times_T T' \to T'$. The fiber over $T$ is the category of schemes over $T$, and pullback is fiber product.

Let $\mathcal{F}$ be the category of vector bundles: objects are pairs $(X, E)$ where $X$ is a smooth manifold (or scheme) and $E \to X$ is a vector bundle. Morphisms are bundle maps covering base maps. The functor $p(X, E) = X$ is fibered, with cartesian morphisms being pullback bundles $f^*E$.

The fiber $\mathcal{F}_X$ is the category $\mathbf{Vect}(X)$ of vector bundles on $X$.

Let $\mathcal{C} = \mathbf{Sch}$ and let $\mathcal{F}$ be the category whose objects are pairs $(S, X \to S)$ where $X \to S$ is a finite etale morphism. This is fibered over $\mathbf{Sch}$: the cartesian lift of $X \to S$ along $T \to S$ is the base change $X \times_S T \to T$, which is again finite etale.

The fiber over $S$ is the category $\mathbf{FEt}_S$ of finite etale covers of $S$. When $S = \operatorname{Spec} k$, this is the category of finite $\operatorname{Gal}(\bar{k}/k)$-sets, central to Grothendieck's algebraic fundamental group.

Fix a group scheme $G$ over a base scheme $S$. Define a category $\mathcal{F}$ whose objects over $T \in \mathbf{Sch}/S$ are principal $G_T$-bundles $P \to T$ (locally trivial in the appropriate topology). A morphism from $(T', P')$ to $(T, P)$ over $f : T' \to T$ is an isomorphism $P' \cong f^*P$.

This is fibered over $\mathbf{Sch}/S$. The fiber over $T$ is the groupoid of principal $G_T$-bundles, which we denote $BG(T)$. This is the classifying stack of $G$.

Every presheaf of categories $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Cat}$ gives rise to a split fibered category via the Grothendieck construction. The total category $\int F$ (also denoted $\mathcal{C} \int F$ or the category of elements) has:

• Objects: pairs $(U, x)$ with $U \in \mathcal{C}$ and $x \in F(U)$
• Morphisms $(V, y) \to (U, x)$: pairs $(f, \phi)$ where $f : V \to U$ in $\mathcal{C}$ and $\phi : y \to F(f)(x)$ in $F(V)$

The projection $(U, x) \mapsto U$ is a split fibered category with splitting $f^*(U, x) = (V, F(f)(x))$.

Consider the fibered category of quasi-coherent sheaves over $\mathbf{Sch}$. Even after choosing a cleavage, the pullback functors generally satisfy $(f \circ g)^* \cong g^* \circ f^*$ only up to natural isomorphism, not strict equality. This is because the pullback $f^*\mathcal{E}$ is typically defined as $\mathcal{O}_{T'} \otimes_{f^{-1}\mathcal{O}_T} f^{-1}\mathcal{E}$, and the tensor product is only associative up to isomorphism.

The associativity isomorphisms $\alpha_{f,g} : g^* \circ f^* \xrightarrow{\sim} (f \circ g)^*$ satisfy a cocycle condition, giving a pseudofunctor (or 2-functor) rather than a strict functor.

The assignment $S \mapsto \mathbf{QCoh}(S)$ (the category of quasi-coherent sheaves on a scheme $S$) defines a pseudofunctor $\mathbf{Sch}^{\mathrm{op}} \to \mathbf{Cat}$. For a morphism $f : T \to S$, the pullback functor is $f^* : \mathbf{QCoh}(S) \to \mathbf{QCoh}(T)$.

The coherence isomorphisms $g^* \circ f^* \cong (f \circ g)^*$ come from the canonical isomorphisms of tensor products. This pseudofunctor corresponds to the fibered category of quasi-coherent sheaves.

Let $f : V \to U$ be a morphism in $\mathcal{C}$ and $x \in \mathcal{F}_U$. If $\phi : y \to x$ and $\phi' : y' \to x$ are both cartesian over $f$, then by the universal property of $\phi$ applied to $\phi'$, there exists a unique $\chi : y' \to y$ in $\mathcal{F}_V$ with $\phi \circ \chi = \phi'$. Symmetrically, there is a unique $\chi' : y \to y'$ with $\phi' \circ \chi' = \phi$. Uniqueness forces $\chi \circ \chi' = \operatorname{id}_y$ and $\chi' \circ \chi = \operatorname{id}_{y'}$. So $y \cong y'$ canonically in $\mathcal{F}_V$.

This is why the pullback $f^*x$ is "well-defined up to unique isomorphism."

Let $W \xrightarrow{g} V \xrightarrow{f} U$ in $\mathcal{C}$ and $x \in \mathcal{F}_U$. Choose cartesian lifts $\phi : f^*x \to x$ over $f$ and $\psi : g^*(f^*x) \to f^*x$ over $g$. The composition $\phi \circ \psi : g^*(f^*x) \to x$ lies over $f \circ g$ and is cartesian.

By uniqueness, $g^*(f^*x) \cong (f \circ g)^*x$ canonically. This canonical isomorphism is the coherence datum $\alpha_{f,g}$ from the pseudofunctor perspective.

Consider the fibered categories $\mathbf{QCoh}$ and $\mathbf{Sh}_{\mathrm{et}}$ over $\mathbf{Sch}$, where $\mathbf{Sh}_{\mathrm{et}}(S)$ is the category of etale sheaves on $S$. The functor sending a quasi-coherent sheaf $\mathcal{E}$ on $S$ to its associated etale sheaf $\mathcal{E}^{\mathrm{et}}$ defines a morphism of fibered categories, since pullback of quasi-coherent sheaves is compatible with the etale sheafification.

Define $\mathcal{M}_g$ as a category over $\mathbf{Sch}$: objects over $S$ are smooth proper morphisms $f : X \to S$ whose geometric fibers are connected curves of genus $g$. A morphism from $(T, Y \to T)$ to $(S, X \to S)$ over $h : T \to S$ is an isomorphism $Y \cong X \times_S T$ over $T$.

This is a fibered category (in fact, fibered in groupoids). The fiber over $\operatorname{Spec} k$ is the groupoid of smooth curves of genus $g$ over $k$. This is the starting point for the moduli stack $\mathcal{M}_g$.

Fix a smooth group scheme $G/S$. Define $\mathcal{B}G$ over $\mathbf{Sch}/S$: objects over $T$ are right $G_T$-torsors $P \to T$ (for the fppf or etale topology). Morphisms are $G$-equivariant isomorphisms compatible with base change.

The fiber over $T$ is the groupoid of $G_T$-torsors. When $G = \mathbb{G}_m$, torsors correspond to line bundles, so $\mathcal{B}\mathbb{G}_m(T) \simeq \operatorname{Pic}(T)$ as a groupoid. When $G = GL_n$, torsors correspond to rank-$n$ vector bundles.