Lemma 101.39.3. In Definition 101.39.1 assume \mathcal{I}_\mathcal {Y} \to \mathcal{Y} is proper (for example if \mathcal{Y} is separated or if \mathcal{Y} is separated over an algebraic space). Then the category of dotted arrows is independent (up to noncanonical equivalence) of the choice of \gamma and the existence of a dotted arrow (for some and hence equivalently all \gamma ) is equivalent to the existence of a diagram
\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y \ar[ru]_ a & \mathcal{Y} }
with 2-commutative triangles (without checking the 2-morphisms compose correctly).
Proof.
Let \gamma , \gamma ' : y \circ j \longrightarrow f \circ x be two 2-morphisms. Then \gamma ^{-1} \circ \gamma ' is an automorphism of y over \mathop{\mathrm{Spec}}(K). Hence if \mathit{Isom}_\mathcal {Y}(y, y) \to \mathop{\mathrm{Spec}}(A) is proper, then by the valuative criterion of properness (Morphisms of Spaces, Lemma 67.44.1) we can find \delta : y \to y whose restriction to \mathop{\mathrm{Spec}}(K) is \gamma ^{-1} \circ \gamma '. Then we can use \delta to define an equivalence between the category of dotted arrows for \gamma to the category of dotted arrows for \gamma ' by sending (a, \alpha , \beta ) to (a, \alpha , \beta \circ \delta ). The final statement is clear.
\square
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