For $\mathcal{X} = \mathcal{M}_g$ (moduli of genus-$g$ curves), the 2-Yoneda lemma gives:

\operatorname{Hom}(h_S, \mathcal{M}_g) \simeq \mathcal{M}_g(S) = \left\{\parbox{20em}{\centering smooth proper morphisms $C \to S$ with geometrically connected fibers of genus $g$}\right\}

A morphism $S \to \mathcal{M}_g$ is a family of curves over $S$. A 2-morphism between two morphisms $f, g : S \to \mathcal{M}_g$ is an isomorphism between the corresponding families. The automorphism group of a morphism $f : S \to \mathcal{M}_g$ is $\operatorname{Aut}(C/S)$, the automorphism group of the corresponding family.

A CFG $\mathcal{X}$ is representable by an object $X \in \mathcal{C}$ if and only if there exists $X$ and an equivalence $\mathcal{X} \simeq h_X$. By the 2-Yoneda lemma, this is equivalent to:

1. $\mathcal{X}(U)$ is a discrete groupoid (set) for all $U$.
2. The functor $U \mapsto \pi_0(\mathcal{X}(U))$ is representable.

For stacks on $(\mathbf{Sch}, \mathrm{fppf})$, condition (1) means all automorphism groups are trivial.

For stacks $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ with morphisms $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$, the 2-fiber product $\mathcal{X} \times_\mathcal{Z} \mathcal{Y}$ is characterized by:

$(\mathcal{X} \times_\mathcal{Z} \mathcal{Y})(U) \simeq \mathcal{X}(U) \times_{\mathcal{Z}(U)} \mathcal{Y}(U)$

where the right-hand side is the 2-fiber product of groupoids. This follows from the 2-Yoneda lemma since $\operatorname{Hom}(h_U, \mathcal{X} \times_\mathcal{Z} \mathcal{Y}) \simeq \operatorname{Hom}(h_U, \mathcal{X}) \times_{\operatorname{Hom}(h_U, \mathcal{Z})} \operatorname{Hom}(h_U, \mathcal{Y})$.

The diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ of a stack $\mathcal{X}$ is representable if and only if for every $U \in \mathcal{C}$ and every pair of objects $x, y \in \mathcal{X}(U)$, the presheaf $\operatorname{Isom}(x, y)$ is representable (by an object of $\mathcal{C}$, or an algebraic space).

By the 2-Yoneda lemma, a morphism $h_U \to \mathcal{X} \times \mathcal{X}$ corresponds to a pair $(x, y) \in \mathcal{X}(U) \times \mathcal{X}(U)$, and: $h_U \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X} \simeq \operatorname{Isom}(x, y)$ So representability of $\Delta$ amounts to representability of all Isom functors.

An atlas (or presentation) of a stack $\mathcal{X}$ is a representable morphism $p : h_U \to \mathcal{X}$ that is a smooth surjection. By the 2-Yoneda lemma, $p$ corresponds to an object $x \in \mathcal{X}(U)$.

The conditions on $p$ translate to:

• Surjective: For every field $k$ and $y \in \mathcal{X}(\operatorname{Spec} k)$, there exists a field extension $k'/k$ and an isomorphism $y|_{k'} \cong x|_{k'}$ in $\mathcal{X}(k')$. (Locally, every object comes from $x$.)
• Smooth: The fiber product $h_U \times_\mathcal{X} h_U = \operatorname{Isom}(x, x) \times \cdots$ (the groupoid scheme $R = U \times_\mathcal{X} U$) is smooth over $U$ via both projections.

For a stack $\mathcal{X}$, the identity morphism $\operatorname{id} : \mathcal{X} \to \mathcal{X}$ corresponds (by a generalized 2-Yoneda) to the universal object over $\mathcal{X}$. For example:

• For $\mathcal{M}_g$: the identity corresponds to the universal curve $\mathcal{C}_g \to \mathcal{M}_g$.
• For $BG$: the identity corresponds to the universal torsor $EG \to BG$ (which is the morphism $\operatorname{Spec} k \to BG$ corresponding to the trivial torsor).
• For $B\mathbb{G}_m$: the identity corresponds to the universal line bundle on $B\mathbb{G}_m$.

These universal objects are constructed abstractly from the 2-Yoneda lemma.

If $\mathcal{C} = \mathbf{Sch}$ and we allow algebraic spaces, then for an algebraic space $X$ (which is a sheaf on the etale site, but not necessarily representable by a scheme):

$\operatorname{Hom}_{\mathbf{Stacks}}(X, \mathcal{F}) \simeq \mathcal{F}(X)$

where $\mathcal{F}(X)$ is defined by extending $\mathcal{F}$ from schemes to algebraic spaces (using the etale presentation $U \to X$ and descent). This allows us to treat algebraic spaces on equal footing with schemes in the functor-of-points language.

In derived algebraic geometry (Lurie, Toen-Vezzosi), the 2-Yoneda lemma generalizes to the $\infty$-Yoneda lemma: for a derived stack $\mathcal{X}$ (a functor $\mathbf{dAff}^{\mathrm{op}} \to \mathcal{S}$ satisfying descent, where $\mathcal{S}$ is the $\infty$-category of spaces):

$\operatorname{Map}(h_{\operatorname{Spec} A}, \mathcal{X}) \simeq \mathcal{X}(\operatorname{Spec} A)$

as an equivalence of spaces (not just groupoids). Here $\operatorname{Map}$ is the mapping space in the $\infty$-category of derived stacks. The classical 2-Yoneda lemma is the truncation to $\pi_0$ and $\pi_1$.

Naturality in $U$ means: if $x \in \mathcal{F}(U)$ corresponds to $\Phi_x : h_U \to \mathcal{F}$, then the pullback $f^*x \in \mathcal{F}(V)$ corresponds to $\Phi_x \circ h_f : h_V \to h_U \to \mathcal{F}$. In moduli terms: if $C \to S$ is a family (corresponding to $S \to \mathcal{M}_g$) and $f : T \to S$, then $f^*C = C \times_S T$ corresponds to the composition $T \to S \to \mathcal{M}_g$.