Lemma 109.23.2. Let $S$ be a scheme and $s \in S$ a point. Let $f : X \to S$ and $g_ i : Y_ i \to S$, $i = 1, 2$ be families of curves. Let $c_ i : X \to Y_ i$ be morphisms over $S$. Assume there is an isomorphism $Y_{1, s} \cong Y_{2, s}$ of fibres compatible with $c_{1, s}$ and $c_{2, s}$. If $c_{1, s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\mathcal{O}_{X_ s} = 0$, then there exist an open neighbourhood $U$ of $s$ and an isomorphism $Y_{1, U} \cong Y_{2, U}$ of families of curves over $U$ compatible with the given isomorphism of fibres and with $c_1$ and $c_2$.
Proof. Recall that $\mathcal{O}_{S, s} = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_ S(U)$ where the colimit is over the system of affine neighbourhoods $U$ of $s$. Thus the category of algebraic spaces of finite presentation over the local ring is the colimit of the categories of algebraic spaces of finite presentation over the affine neighbourhoods of $s$. See Limits of Spaces, Lemma 70.7.1. In this way we reduce to the case where $S$ is the spectrum of a local ring and $s$ is the closed point.
Assume $S = \mathop{\mathrm{Spec}}(A)$ where $A$ is a local ring and $s$ is the closed point. Write $A = \mathop{\mathrm{colim}}\nolimits A_ j$ with $A_ j$ local Noetherian (say essentially of finite type over $\mathbf{Z}$) and local transition homomorphisms. Set $S_ j = \mathop{\mathrm{Spec}}(A_ j)$ with closed point $s_ j$. We can find a $j$ and families of curves $X_ j \to S_ j$, $Y_{j, i} \to S_ j$, see Lemma 109.5.3 and Limits of Stacks, Lemma 102.3.5. After possibly increasing $j$ we can find morphisms $c_{j, i} : X_ j \to Y_{j, i}$ whose base change to $s$ is $c_ i$, see Limits of Spaces, Lemma 70.7.1. Since $\kappa (s) = \mathop{\mathrm{colim}}\nolimits \kappa (s_ j)$ we can similarly assume there is an isomorphism $Y_{j, 1, s_ j} \cong Y_{j, 2, s_ j}$ compatible with $c_{j, 1, s_ j}$ and $c_{j, 2, s_ j}$. Finally, the assumptions $c_{1, s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\mathcal{O}_{X_ s} = 0$ are inherited by $c_{j, 1, s_ j}$ because $\{ s_ j \to s\} $ is an fpqc covering and $c_{1, s}$ is the base of $c_{j, 1, s_ j}$ by this covering (details omitted). In this way we reduce the lemma to the case discussed in the next paragraph.
Assume $S$ is the spectrum of a Noetherian local ring $\Lambda $ and $s$ is the closed point. Consider the scheme theoretic image $Z$ of
\[ (c_1, c_2) : X \longrightarrow Y_1 \times _ S Y_2 \]
The statement of the lemma is equivalent to the assertion that $Z$ maps isomorphically to $Y_1$ and $Y_2$ via the projection morphisms. Since taking the scheme theoretic image of this morphism commutes with flat base change (Morphisms of Spaces, Lemma 67.30.12, we may replace $\Lambda $ by its completion (More on Algebra, Section 15.43).
Assume $S$ is the spectrum of a complete Noetherian local ring $\Lambda $. Observe that $X$, $Y_1$, $Y_2$ are schemes in this case (More on Morphisms of Spaces, Lemma 76.43.6). Denote $X_ n$, $Y_{1, n}$, $Y_{2, n}$ the base changes of $X$, $Y_1$, $Y_2$ to $\mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^{n + 1})$. Recall that the arrow
\[ \mathcal{D}\! \mathit{ef}_{X_ s \to Y_{2, s}} \cong \mathcal{D}\! \mathit{ef}_{X_ s \to Y_{1, s}} \longrightarrow \mathcal{D}\! \mathit{ef}_{X_ s} \]
is an equivalence, see Deformation Problems, Lemma 93.10.6. Thus there is an isomorphism of formal objects $(X_ n \to Y_{1, n}) \cong (X_ n \to Y_{2, n})$ of $\mathcal{D}\! \mathit{ef}_{X_ s \to Y_{1, s}}$. Finally, by Grothendieck's algebraization theorem (Cohomology of Schemes, Lemma 30.28.3) this produces an isomorphism $Y_1 \to Y_2$ compatible with $c_1$ and $c_2$. $\square$
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