preservesFiniteLimits — Mathlib

English

If F: D → C^op is a functor with nonempty D, then the embedding of F is faithful.

Русский

Если F: D → C^op — функтор с непустым D, то вложение F достоверно (faithful).

LaTeX

$$F . embedding is faithful$$

Lean4

/-- Right exactness follows because `tensorObj G` is a left adjoint. -/
theorem preservesFiniteLimits (G : C) (hG : IsSeparator G) : PreservesFiniteLimits (tensorObj G) :=
  by
  have := preservesInjectiveObjects G hG
  have : (tensorObj G).PreservesMonomorphisms :=
    (tensorObj G).preservesMonomorphisms_of_adjunction_of_preservesInjectiveObjects
      (tensorObjPreadditiveCoyonedaObjAdjunction G)
  have : PreservesBinaryBiproducts (tensorObj G) := preservesBinaryBiproducts_of_preservesBinaryCoproducts _
  have : (tensorObj G).Additive := Functor.additive_of_preservesBinaryBiproducts _
  have : (tensorObj G).PreservesHomology := (tensorObj G).preservesHomology_of_preservesMonos_and_cokernels
  exact (tensorObj G).preservesFiniteLimits_of_preservesHomology

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