This is something that has nagged me for a while, and I think I basically know the issue, but thought I would see what people had to say.
In the study of moduli of curves, one will sometimes see the following morphism referred to as the "universal family" or the "universal genus $g$ curve":
$$\pi : M_{g,1} \to M_{g}.$$
Here $M_{g}$ is the coarse moduli space of smooth genus $g$ curves, and $M_{g,1}$ is the moduli space of smooth genus $g$ curves with one marked point. It's pretty clear intuitively why $\pi$ above would be a candidate for the universal family, but we know the moduli space of smooth curves is not fine! So equivalently, there is no universal family.
I am under the impression that this is basically an abuse of terminology. In other words, $\pi: M_{g,1} \to M_{g}$ is not THE universal family—there are non-trivial families of curves with automorphisms that do not pull back from $\pi$. But it is a family where every curve appears once, so people via an abusive of terminology call it the universal family.
Is my idea here correct? If so, is this something to be wary of in moduli theory–people speaking about universal families despite the moduli problem not being representable?
Best Answer
This is really categorical nonsense to spell out.
$ \mathcal{M}_{g,1} $ is the category fibered in groupoids (cfig) over schemes over a field $ k $ whose objects consist of families $ \pi : X \rightarrow S $ where $ \pi $ is smooth proper with every geometric fiber a nonsingular genus $ g $ curve, along with a section $ s $ of $ \pi $. The morphisms are given by obvious commutative diagrams. $ \mathcal{M}_g $ is made up of the same data except the section. The morphism $ \mathcal{M}_{g,1} \rightarrow \mathcal{M}_g $ is also the obvious one, given by forgetting the section.
To see that this is the universal family, we need to show that every family in $ \mathcal{M}_g $ arises by pullback. So consider a family $ \pi : X \rightarrow S $ of curves as above (no section here). By the 2-Yoneda Lemma, there is a unique morphism $ h_{\pi} : \underline{S} \rightarrow \mathcal{M}_g $ inducing $ \pi $, where $ \underline{S} $ is the cfig associated to $ S $.
We claim that the fiber product $ \underline{S} \times_{\mathcal{M}_g} \mathcal{M}_{g,1} $ is $ \underline{X} $. By definition, an object of the fiber product is given by a compatible triple (over the same object $ T $ in the base category of schemes over $ k $), which means the following data:
(1) $ (T,g) $ an object of $ \underline{S} $, i.e. $ g : T \rightarrow S $ a morphism.
(2) $ \eta : Y \rightarrow T $ a family in $ \mathcal{M}_{g,1} (T) $ so that it comes with a section $ s $.
(3) $ \gamma $ an isomorphism given by $ \gamma : Y \rightarrow T \times_S X $ over $ T $.
The morphisms between objects in the fiber product are given by the obvious diagrams again.
Points (2) and (3) together imply that $ T \times_S X \rightarrow T $ has a section and giving a section of this map is equivalent to giving a map $ f : T \rightarrow X $ such that $ \pi \circ f = g $. All of this is clearly reversible: every map $ T \rightarrow X $ gives a compatible triple in the fiber product.
In summary, to specify the objects of the fiber product over $ T $ is equivalent to specifying morphisms $ T \rightarrow X $. So we proved that the fiber product is precisely $ \underline{X} $ and the morphism $ \underline{X} \rightarrow \underline{S} $ is $ \underline{\pi} $ by construction, showing that $ \mathcal{M}_{g,1} \rightarrow \mathcal{M}_g $ is indeed universal.
And yes, to answer one of your original questions, the moduli functor of smooth curves is not representable in the world of schemes, i.e. there does not exist a universal family (for the reasons of automorphisms as you allude to). The coarse moduli space just parametizes curves in the sense of closed points corresponding to curves. It doesn't come with a universal family.