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$g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{C} \to \mathcal{C}'$ (see Sites, Definition 7.29.2),

Corporation Tax Act 2010, Section 1046 is up to date with all changes known to be in force on or before 24 October 2024. There are changes that may be brought into force at a future date. Changes that have been made appear in the content and are referenced with annotations.

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F1S. 1046(5)-(7) omitted (with effect in accordance with Sch. 23 para. 65 of the amending Act) by virtue of Finance Act 2011 (c. 11), Sch. 23 paras. 64(2)(d), 65(1)(a) (with Sch. 23 paras. 50, 65(1)(b))

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given any set of sheaves $\mathcal{F}_ i$ (resp. $\mathcal{G}_ j$) on $\mathcal{C}$ (resp. $\mathcal{D}$) we may assume each of these is a representable sheaf on $\mathcal{C}'$ (resp. $\mathcal{D}'$).

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Lemma 18.7.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. There exists a factorization

$g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{C} \to \mathcal{C}'$ (see Sites, Definition 7.29.2), $e : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{D} \to \mathcal{D}'$ (see Sites, Definition 7.29.2), $\mathcal{O}_{\mathcal{C}'} = g_*\mathcal{O}_\mathcal {C}$ and $g^\sharp $ is the obvious map, $\mathcal{O}_{\mathcal{D}'} = e_*\mathcal{O}_\mathcal {D}$ and $e^\sharp $ is the obvious map, the sites $\mathcal{C}'$ and $\mathcal{D}'$ have final objects and fibre products (i.e., all finite limits), $h$ is a morphism of sites induced by a continuous functor $u : \mathcal{D}' \to \mathcal{C}'$ which commutes with all finite limits (i.e., it satisfies the assumptions of Sites, Proposition 7.14.7), and given any set of sheaves $\mathcal{F}_ i$ (resp. $\mathcal{G}_ j$) on $\mathcal{C}$ (resp. $\mathcal{D}$) we may assume each of these is a representable sheaf on $\mathcal{C}'$ (resp. $\mathcal{D}'$).

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$e : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{D} \to \mathcal{D}'$ (see Sites, Definition 7.29.2),

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(a)the company treats a payment made by it as one to which section 1033 applies and in relation to which Condition A in that section is met, and

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(1)A company which treats a payment made by it as one to which section 1033 applies must make a return to an officer of Revenue and Customs giving details of—

Moreover, if $(f, f^\sharp )$ is an equivalence of ringed topoi, then we can choose the diagram such that $\mathcal{C}' = \mathcal{D}'$, $\mathcal{O}_{\mathcal{C}'} = \mathcal{O}_{\mathcal{D}'}$ and $(h, h^\sharp )$ is the identity.

Proof. This follows from Sites, Lemma 7.29.6, and Sites, Remarks 7.29.7 and 7.29.8. You just have to carry along the sheaves of rings. Some details omitted. $\square$

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$h$ is a morphism of sites induced by a continuous functor $u : \mathcal{D}' \to \mathcal{C}'$ which commutes with all finite limits (i.e., it satisfies the assumptions of Sites, Proposition 7.14.7), and