English
In the over category with a pullback and a binary product, the forward morphism interacts with the pullback’s fst projection through the left iso in a precise way: the composed morphism equals the left projection of the product.
Русский
В категории Over при наличии gij-пересечения и бинарного произведения композиции правого стрелы через левый изоморфизм равны левой проекции произведения.
LaTeX
$$$$ (\mathrm{prodLeftIsoPullback}\, Y\, Z)^{\mathrm{hom}} \circ \mathrm{pullback.fst}\; (Y.hom, Z.hom) = \mathrm{prod.fst}_{X:=Y}.\mathrm{left} $$$$
Lean4
/-- The forgetful functor from `CostructuredArrow K B` preserves any connected limit. -/
instance [IsConnected J] {B : D} : PreservesLimitsOfShape J (CostructuredArrow.proj K B) where
preservesLimit.preserves
hc :=
⟨{ lift s := (CostructuredArrow.proj K B).map (hc.lift (CreatesConnected.raiseCone s))
fac _
_ := by
rw [Functor.mapCone_π_app, ← Functor.map_comp, hc.fac, CreatesConnected.raiseCone_π_app,
CostructuredArrow.proj_map, CostructuredArrow.homMk_left _ _]
uniq s m
fac :=
congrArg (CostructuredArrow.proj K B).map
(hc.uniq (CreatesConnected.raiseCone s) (CostructuredArrow.homMk m (by simp [← fac])) fun j =>
(CostructuredArrow.proj K B).map_injective (fac j)) }⟩
