Theorem 99.15.11 (Algebraicity of the stack of curves). The stack $\mathcal{C}\! \mathit{urves}$ (Situation 99.15.1) is algebraic. In fact, for any algebraic space $B$ the stack $B\text{-}\mathcal{C}\! \mathit{urves}$ (Remark 99.15.5) is algebraic.
See [Proposition 3.3, page 8, dJHS] and [Appendix B by Jack Hall, Theorem B.1, Smyth].
Proof.
The absolute case follows from Artin's Axioms, Lemma 98.17.1 and Lemmas 99.15.4, 99.15.7, 99.15.6, 99.15.9, and 99.15.10. The case over $B$ follows from this, the description of $B\text{-}\mathcal{C}\! \mathit{urves}$ as a $2$-fibre product in Remark 99.15.5, and the fact that algebraic stacks have $2$-fibre products, see Algebraic Stacks, Lemma 94.14.3.
$\square$
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