-- Mathlib/Condensed/Discrete/Characterization.lean
theorem isDiscrete_tfae  (M : LightCondMod.{u} R) :
    TFAE
    [ M.IsDiscrete
    , IsIso ((LightCondensed.discreteUnderlyingAdj _).counit.app M)
    , (LightCondensed.discrete _).essImage M
    , (LightCondMod.LocallyConstant.functor R).essImage M
    , IsIso ((LightCondMod.LocallyConstant.adjunction R).counit.app M)
    , ∀ S : LightProfinite.{u}, Nonempty
        (IsColimit <| M.val.mapCocone (coconeRightOpOfCone S.asLimitCone))
    ]

Proof state:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
⊢ [LightCondensed.IsDiscrete M, IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
      (LightCondensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
      IsIso ((LocallyConstant.adjunction R).counit.app M),
      ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))].TFAE

Tactic:
tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _
  LightProfinite.isTerminalPUnit (adjunction R) _

Result:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
tfae_1_iff_4 : LightCondensed.IsDiscrete M ↔ (functor R).essImage M
⊢ [LightCondensed.IsDiscrete M, IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
      (LightCondensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
      IsIso ((LocallyConstant.adjunction R).counit.app M),
      ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))].TFAE

Proof state:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
⊢ [LightCondensed.IsDiscrete M, IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
      (LightCondensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
      IsIso ((LocallyConstant.adjunction R).counit.app M),
      ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))].TFAE

Tactic:
tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩

Result:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
⊢ [LightCondensed.IsDiscrete M, IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
      (LightCondensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
      IsIso ((LocallyConstant.adjunction R).counit.app M),
      ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))].TFAE

Proof state:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
tfae_1_iff_4 : LightCondensed.IsDiscrete M ↔ (functor R).essImage M
⊢ [LightCondensed.IsDiscrete M, IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
      (LightCondensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
      IsIso ((LocallyConstant.adjunction R).counit.app M),
      ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))].TFAE

Tactic:
tfae_have 1 ↔ 5 :=
  have : (functor R).Faithful := inferInstance
  have : (functor R).Full := inferInstance
  Sheaf.isConstant_iff_isIso_counit_app' _ LightProfinite.isTerminalPUnit (adjunction R) _

Result:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
tfae_1_iff_4 : LightCondensed.IsDiscrete M ↔ (functor R).essImage M
tfae_1_iff_5 : LightCondensed.IsDiscrete M ↔ IsIso ((LocallyConstant.adjunction R).counit.app M)
⊢ [LightCondensed.IsDiscrete M, IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M),
      (LightCondensed.discrete (ModuleCat R)).essImage M, (functor R).essImage M,
      IsIso ((LocallyConstant.adjunction R).counit.app M),
      ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))].TFAE

Proof state:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
tfae_1_iff_4 : LightCondensed.IsDiscrete M ↔ (functor R).essImage M
tfae_1_iff_5 : LightCondensed.IsDiscrete M ↔ IsIso ((LocallyConstant.adjunction R).counit.app M)
tfae_6_to_1 :
  (∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))) →
    LightCondensed.IsDiscrete M
⊢ LightCondensed.IsDiscrete M →
    ∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))

Tactic:
intro h S

Result:
R : Type u
inst✝ : Ring R
M : LightCondMod R
tfae_1_iff_2 : LightCondensed.IsDiscrete M ↔ IsIso ((LightCondensed.discreteUnderlyingAdj (ModuleCat R)).counit.app M)
tfae_1_iff_3 : LightCondensed.IsDiscrete M ↔ (LightCondensed.discrete (ModuleCat R)).essImage M
tfae_1_iff_4 : LightCondensed.IsDiscrete M ↔ (functor R).essImage M
tfae_1_iff_5 : LightCondensed.IsDiscrete M ↔ IsIso ((LocallyConstant.adjunction R).counit.app M)
tfae_6_to_1 :
  (∀ (S : LightProfinite), Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))) →
    LightCondensed.IsDiscrete M
h : LightCondensed.IsDiscrete M
S : LightProfinite
⊢ Nonempty (IsColimit (M.val.mapCocone (coconeRightOpOfCone S.asLimitCone)))