I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack of elliptic curves. Then, they declare that $S \to S' \to \mathcal{M}_{1, 1}$ is commutative (there is an extra map $S \to \mathcal{M}_{1, 1}$ but I can't fit in since I don't know how to put commutative diagram here) when there is a morphism of elliptic curves $E \to E'$ over $S \to S'$. I think we should really add an extra assumption that the induced map $E \to E' \times_{S'} S$ is an isomorphism. At least, this would make sense when $\mathcal{M}_{1, 1}$ is a scheme and there exists a universal family of elliptic curves.

I hope someone can illuminate me with this. Thanks!