Lemma 66.28.2. Let S be a scheme. Let X, Y be algebraic spaces over S. Let f : X \to Y be a morphism of algebraic spaces over S. Let t be a 2-morphism from (f_{small}, f^\sharp ) to itself, see Modules on Sites, Definition 18.8.1. Then t = \text{id}.
Proof. Let X', resp. Y' be X viewed as an algebraic space over \mathop{\mathrm{Spec}}(\mathbf{Z}), see Spaces, Definition 65.16.2. It is clear from the construction that (X_{small}, \mathcal{O}) is equal to (X'_{small}, \mathcal{O}) and similarly for Y. Hence we may work with X' and Y'. In other words we may assume that S = \mathop{\mathrm{Spec}}(\mathbf{Z}).
Assume S = \mathop{\mathrm{Spec}}(\mathbf{Z}), f : X \to Y and t are as in the lemma. This means that t : f^{-1}_{small} \to f^{-1}_{small} is a transformation of functors such that the diagram
\xymatrix{ f_{small}^{-1}\mathcal{O}_ Y \ar[rd]_{f^\sharp } & & f_{small}^{-1}\mathcal{O}_ Y \ar[ll]^ t \ar[ld]^{f^\sharp } \\ & \mathcal{O}_ X }
is commutative. Suppose V \to Y is étale with V affine. Write V = \mathop{\mathrm{Spec}}(B). Choose generators b_ j \in B, j \in J for B as a \mathbf{Z}-algebra. Set T = \mathop{\mathrm{Spec}}(\mathbf{Z}[\{ x_ j\} _{j \in J}]). In the following we will use that \mathop{\mathrm{Mor}}\nolimits _{\mathit{Sch}}(U, T) = \prod _{j \in J} \Gamma (U, \mathcal{O}_ U) for any scheme U without further mention. The surjective ring map \mathbf{Z}[x_ j] \to B, x_ j \mapsto b_ j corresponds to a closed immersion V \to T. We obtain a monomorphism
i : V \longrightarrow T_ Y = T \times Y
of algebraic spaces over Y. In terms of sheaves on Y_{\acute{e}tale} the morphism i induces an injection h_ i : h_ V \to \prod _{j \in J} \mathcal{O}_ Y of sheaves. The base change i' : X \times _ Y V \to T_ X of i to X is a monomorphism too (Spaces, Lemma 65.5.5). Hence i' : X \times _ Y V \to T_ X is a monomorphism, which in turn means that h_{i'} : h_{X \times _ Y V} \to \prod _{j \in J} \mathcal{O}_ X is an injection of sheaves. Via the identification f_{small}^{-1}h_ V = h_{X \times _ Y V} of Lemma 66.19.9 the map h_{i'} is equal to
\xymatrix{ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^{\prod f^\sharp } & \prod _{j \in J} \mathcal{O}_ X }
(verification omitted). This means that the map t : f_{small}^{-1}h_ V \to f_{small}^{-1}h_ V fits into the commutative diagram
\xymatrix{ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} \ar[d]^ t & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^-{\prod f^\sharp } \ar[d]^{\prod t} & \prod _{j \in J} \mathcal{O}_ X \ar[d]^{\text{id}}\\ f_{small}^{-1}h_ V \ar[r]^-{f^{-1}h_ i} & \prod _{j \in J} f_{small}^{-1}\mathcal{O}_ Y \ar[r]^-{\prod f^\sharp } & \prod _{j \in J} \mathcal{O}_ X }
The commutativity of the right square holds by our assumption on t explained above. Since the composition of the horizontal arrows is injective by the discussion above we conclude that the left vertical arrow is the identity map as well. Any sheaf of sets on Y_{\acute{e}tale} admits a surjection from a (huge) coproduct of sheaves of the form h_ V with V affine (combine Lemma 66.18.6 with Sites, Lemma 7.12.5). Thus we conclude that t : f_{small}^{-1} \to f_{small}^{-1} is the identity transformation as desired. \square
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