English
If G preserves finite coproducts, then the Yoneda presheaf along G preserves finite products; this is used for the extensive topology.
Русский
Если G сохраняет конечные диепредельники, тогда Yoneda-пред_presheaf вдоль G сохраняет конечные произведения; это применяется к экстентному топологическому vybavaniju.
LaTeX
$$$\text{PreservesFiniteProducts}(\mathrm{yonedaPresheaf}(G,X))$ given $\mathrm{PreservesFiniteCoproducts}(G)$$$
Lean4
/-- If `G` preserves finite coproducts (which is the case when `C` is `CompHaus`, `Profinite` or
`Stonean`), then `yonedaPresheaf` preserves finite products, which is required to be a sheaf for
the extensive topology.
-/
noncomputable instance [PreservesFiniteCoproducts G] : PreservesFiniteProducts (yonedaPresheaf G X) :=
have := preservesFiniteProducts_op G
⟨fun _ ↦ comp_preservesLimitsOfShape G.op (yonedaPresheaf' X)⟩
