English
In the fork diagram associated to a presieve, the two canonical maps commute with the left/right projections and pullback structure.
Русский
В форке, соответствующем прешейву, две канонические стрелы сходятся и согласованы с левым/правым проекциям и структурой выемки.
LaTeX
$$$forkMap \\;\\; ,\\; firstMap,\\; secondMap \\;\\Rightarrow\\; (forkMap \\; \\gg firstMap) = (forkMap \\; \\gg secondMap).$$$
Lean4
instance sheafHomHasZSMul : SMul ℤ (P ⟶ Q) where
smul n
f :=
Sheaf.Hom.mk
{ app := fun U => n • f.1.app U
naturality := fun U V i => by
induction n with
| zero => simp only [zero_smul, comp_zero, zero_comp]
| succ n ih => simpa only [add_zsmul, one_zsmul, comp_add, NatTrans.naturality, add_comp, add_left_inj]
| pred n ih =>
simpa only [sub_smul, one_zsmul, comp_sub, NatTrans.naturality, sub_comp, sub_left_inj] using ih }
