Proof.
Let U = \mathop{\mathrm{lim}}\nolimits U_ i be a directed limit of affine schemes over S. We claim that the functor
\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}
is fully faithful if and only if the functor
\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U} \mathop{\mathrm{colim}}\nolimits (\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i}
is an equivalence. This will prove the lemma. Since (\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{X}_ U and (\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i} = \mathcal{X}_{U_ i} \times _{\mathcal{Y}_{U_ i}} \mathcal{X}_{U_ i} this is a purely category theoretic assertion which we discuss in the next paragraph.
Let \mathcal{I} be a filtered index category. Let (\mathcal{C}_ i) and (\mathcal{D}_ i) be systems of groupoids over \mathcal{I}. Let p : (\mathcal{C}_ i) \to (\mathcal{D}_ i) be a map of systems of groupoids over \mathcal{I}. Suppose we have a functor p : \mathcal{C} \to \mathcal{D} of groupoids and functors f : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \to \mathcal{C} and g : \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \to \mathcal{D} fitting into a commutative diagram
\xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \ar[r]^ g & \mathcal{D} }
Then we claim that
A : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _\mathcal {D} \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i
is fully faithful if and only if the functor
B : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} \mathop{\mathrm{colim}}\nolimits (\mathcal{C}_ i \times _{\mathcal{D}_ i} \mathcal{C}_ i)
is an equivalence. Set \mathcal{C}' = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i and \mathcal{D}' = \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i. Since 2-fibre products commute with filtered colimits we see that A and B become the functors
A' : \mathcal{C}' \to \mathcal{C} \times _\mathcal {D} \mathcal{D}' \quad \text{and}\quad B' : \mathcal{C}' \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}')
Thus it suffices to prove that if
\xymatrix{ \mathcal{C}' \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathcal{D}' \ar[r]^ g & \mathcal{D} }
is a commutative diagram of groupoids, then A' is fully faithful if and only if B' is an equivalence. This follows from Categories, Lemma 4.35.10 (with trivial, i.e., punctual, base category) because
\mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}') = \mathcal{C}' \times _{A', \mathcal{C} \times _\mathcal {D} \mathcal{D}', A'} \mathcal{C}'
This finishes the proof.
\square
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