coneOfHom_fac — Mathlib

English

If cones are representable by X, then the cone obtained via homOfCone and coneOfHom coincides with the limit cone obtained via limitCone and homOfCone.

Русский

Если конусы представлены X, то конус, полученный через homOfCone и coneOfHom, совпадает с предел-конусом, полученным через limitCone и homOfCone.

LaTeX

$$$ coneOfHom_homOfCone = limitCone.homOfCone $$$

Lean4

/-- If `F.cones` is represented by `X`, the cone corresponding to a morphism `f : Y ⟶ X` is
the limit cone extended by `f`. -/
theorem coneOfHom_fac {Y : C} (f : Y ⟶ X) : coneOfHom h f = (limitCone h).extend f :=
  by
  dsimp [coneOfHom, limitCone, Cone.extend]
  congr
  conv_lhs => rw [← Category.comp_id f]
  exact h.homEquiv_comp f (𝟙 X)

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