Let $L/K$ be a field extension, and let $\mathcal{M}$ be some moduli stack (for example, the stack of genus $g$ curves).

Let $X,X'$ be two objects of $\mathcal{M}$ over $K$, giving us two morphisms $X,X':\text{Spec }K\rightarrow\mathcal{M}$. Suppose their pullbacks $X_L,X'_L$ are isomorphic, which is to say that the two composed morphisms $$\text{Spec }L\rightarrow\text{Spec K}\rightrightarrows \mathcal{M}$$ are 2-isomorphic. Now, I sort of want to say that because $\text{Spec }L\rightarrow\text{Spec }K$ is an epimorphism, that $X,X'$ must have determined 2-isomorphic morphisms, and hence were already isomorphic (over $K$) in the first place. ...But this is obviously wrong (for example, take $\mathcal{M}$ to be the moduli stack of elliptic curves, and $X,X'$ to be two nonisomorphic (over $K$) elliptic curves with the same $j$-invariant.)

Where exactly is the problem?

For example,

- Is $p : \text{Spec }L\rightarrow\text{Spec }K$ not an epimorphism in the category of algebraic stacks?
- Perhaps the right question is - Is $p$ a 2-epimorphism in the 2-category of algebraic stacks? What is a down-to-earth definition of a 2-epimorphism anyway? (nlab was not especially helpful in this regard).
- Does the problem that arises in this situation disappear if we assume that $L,K$ are both algebraically closed?

regularepimorphism, hence K --> L to be an effective monomorphism. I'm not sure about such things in algebra... $\endgroup$ – David Roberts Aug 11 '16 at 5:41