diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean
index 451ab44313d2a..f458281e6a6d5 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean
@@ -18,7 +18,7 @@ This file characterises sheaves for the extensive topology.
   extensive topology are precisely those preserving finite products.
 -/
 
-universe v u
+universe v u w
 
 namespace CategoryTheory
 
@@ -49,7 +49,7 @@ A finite product preserving presheaf is a sheaf for the extensive topology on a
 `FinitaryPreExtensive`.
 -/
 theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.Extensive]
-    (F : Cᵒᵖ ⥤ Type max u v) [PreservesFiniteProducts F] : S.IsSheafFor F  := by
+    (F : Cᵒᵖ ⥤ Type w) [PreservesFiniteProducts F] : S.IsSheafFor F  := by
   obtain ⟨α, _, Z, π, rfl, ⟨hc⟩⟩ := Extensive.arrows_nonempty_isColimit (R := S)
   have : (ofArrows Z (Cofan.mk X π).inj).hasPullbacks :=
     (inferInstance : (ofArrows Z π).hasPullbacks)
@@ -62,7 +62,7 @@ instance {α : Type} [Finite α] (Z : α → C) : (ofArrows Z (fun i ↦ Sigma.
 /--
 A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
 -/
-theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type max u v) :
+theorem isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type w) :
     Presieve.IsSheaf (extensiveCoverage C).toGrothendieck F ↔
     Nonempty (PreservesFiniteProducts F) := by
   refine ⟨fun hF ↦ ⟨⟨fun α _ ↦ ⟨fun {K} ↦ ?_⟩⟩⟩, fun hF ↦ ?_⟩
diff --git a/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean b/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
index 942f2ce37e64d..9378029eda4a0 100644
--- a/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
+++ b/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
@@ -263,9 +263,11 @@ theorem sheaf_condition : R.IsSheafFor P ↔ Nonempty (IsLimit (Fork.ofι _ (w P
 
 namespace Arrows
 
+variable (P : Cᵒᵖ ⥤ Type w) {X : C} (R : Presieve X) (S : Sieve X)
+
 open Presieve
 
-variable {B : C} {I : Type} (X : I → C) (π : (i : I) → X i ⟶ B) [UnivLE.{w, max v u}]
+variable {B : C} {I : Type} (X : I → C) (π : (i : I) → X i ⟶ B)
     [(Presieve.ofArrows X π).hasPullbacks]
 -- TODO: allow `I : Type w` 
 
@@ -274,7 +276,7 @@ The middle object of the fork diagram of <https://stacks.math.columbia.edu/tag/0
 The difference between this and `Equalizer.FirstObj P (ofArrows X π)` arrises if the family of
 arrows `π` contains duplicates. The `Presieve.ofArrows` doesn't see those.
 -/
-def FirstObj : Type max v u := ∏ (fun i ↦ P.obj (op (X i)))
+def FirstObj : Type w := ∏ (fun i ↦ P.obj (op (X i)))
 
 @[ext]
 lemma FirstObj.ext (z₁ z₂ : FirstObj P X) (h : ∀ i, (Pi.π _ i : FirstObj P X ⟶ _) z₁ =
@@ -288,7 +290,7 @@ The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag
 The difference between this and `Equalizer.Presieve.SecondObj P (ofArrows X π)` arrises if the
 family of arrows `π` contains duplicates. The `Presieve.ofArrows` doesn't see those.
 -/
-def SecondObj : Type max v u  :=
+def SecondObj : Type w  :=
   ∏ (fun (ij : I × I) ↦ P.obj (op (pullback (π ij.1) (π ij.2))))
 
 @[ext]
diff --git a/Mathlib/CategoryTheory/Sites/Preserves.lean b/Mathlib/CategoryTheory/Sites/Preserves.lean
index db3579853c38c..f9037f2a52da3 100644
--- a/Mathlib/CategoryTheory/Sites/Preserves.lean
+++ b/Mathlib/CategoryTheory/Sites/Preserves.lean
@@ -30,11 +30,11 @@ See `preservesProductOfIsSheafFor`.
 See `isSheafFor_of_preservesProduct`.
 -/
 
-universe v u
+universe v u w
 
 namespace CategoryTheory.Presieve
 
-variable {C : Type u} [Category.{v} C] {I : C} (F : Cᵒᵖ ⥤ Type (max u v))
+variable {C : Type u} [Category.{v} C] {I : C} (F : Cᵒᵖ ⥤ Type w)
 
 open Limits Opposite
 
diff --git a/Mathlib/Condensed/Explicit.lean b/Mathlib/Condensed/Explicit.lean
index 87e5beffb0047..32711ba029a10 100644
--- a/Mathlib/Condensed/Explicit.lean
+++ b/Mathlib/Condensed/Explicit.lean
@@ -27,7 +27,7 @@ continuous surjection), the presheaf `F` exhibits `F(B)` as the equalizer of th
 `F(X) ⇉ F(X ×_B X)`
 -/
 
-universe v u
+universe v u w
 
 open CategoryTheory Limits Opposite Functor Presheaf regularTopology
 
@@ -46,7 +46,7 @@ end CategoryTheory
 namespace CompHaus
 
 theorem isSheaf_iff_preservesFiniteProducts_and_equalizerCondition
-    (F : CompHaus.{u}ᵒᵖ ⥤ Type (u+1)) :
+    (F : CompHaus.{u}ᵒᵖ ⥤ Type w) :
     IsSheaf (coherentTopology CompHaus) F ↔
     Nonempty (PreservesFiniteProducts F) ∧ EqualizerCondition F := by
   rw [isSheaf_coherent_iff_regular_and_extensive]
@@ -55,11 +55,12 @@ theorem isSheaf_iff_preservesFiniteProducts_and_equalizerCondition
   · rw [equalizerCondition_iff_isSheaf, isSheaf_iff_isSheaf_of_type]
 
 theorem isSheaf_iff_preservesFiniteProducts_and_equalizerCondition'
-    {A : Type (u+2)} [Category.{u+1} A] (G : A ⥤ Type (u+1))
-    [HasLimits A] [PreservesLimits G] [G.ReflectsIsomorphisms] (F : CompHaus.{u}ᵒᵖ ⥤ A) :
+    {A : Type*} [Category A] (G : A ⥤ Type w)
+    [HasLimitsOfSize.{u, u+1} A] [PreservesLimitsOfSize.{u, u+1} G]
+    [G.ReflectsIsomorphisms] (F : CompHaus.{u}ᵒᵖ ⥤ A) :
     Presheaf.IsSheaf (coherentTopology CompHaus) F ↔
     Nonempty (PreservesFiniteProducts (F ⋙ G)) ∧ EqualizerCondition (F ⋙ G) := by
-  rw [Presheaf.isSheaf_iff_isSheaf_forget (coherentTopology CompHaus) F G,
+  rw [Presheaf.isSheaf_iff_isSheaf_comp (coherentTopology CompHaus) F G,
     isSheaf_iff_preservesFiniteProducts_and_equalizerCondition]
 
 end CompHaus
@@ -67,7 +68,7 @@ end CompHaus
 namespace Profinite
 
 theorem isSheaf_iff_preservesFiniteProducts_and_equalizerCondition
-    (F : Profinite.{u}ᵒᵖ ⥤ Type (u+1)) :
+    (F : Profinite.{u}ᵒᵖ ⥤ Type w) :
     IsSheaf (coherentTopology Profinite) F ↔
     Nonempty (PreservesFiniteProducts F) ∧ EqualizerCondition F := by
   rw [isSheaf_coherent_iff_regular_and_extensive]
@@ -76,11 +77,12 @@ theorem isSheaf_iff_preservesFiniteProducts_and_equalizerCondition
   · rw [equalizerCondition_iff_isSheaf, isSheaf_iff_isSheaf_of_type]
 
 theorem isSheaf_iff_preservesFiniteProducts_and_equalizerCondition'
-    {A : Type (u+2)} [Category.{u+1} A] (G : A ⥤ Type (u+1))
-    [HasLimits A] [PreservesLimits G] [G.ReflectsIsomorphisms] (F : Profinite.{u}ᵒᵖ ⥤ A) :
+    {A : Type*} [Category A] (G : A ⥤ Type w)
+    [HasLimitsOfSize.{u, u+1} A] [PreservesLimitsOfSize.{u, u+1} G]
+    [G.ReflectsIsomorphisms] (F : Profinite.{u}ᵒᵖ ⥤ A) :
     Presheaf.IsSheaf (coherentTopology Profinite) F ↔
     Nonempty (PreservesFiniteProducts (F ⋙ G)) ∧ EqualizerCondition (F ⋙ G) := by
-  rw [Presheaf.isSheaf_iff_isSheaf_forget (coherentTopology Profinite) F G,
+  rw [Presheaf.isSheaf_iff_isSheaf_comp (coherentTopology Profinite) F G,
     isSheaf_iff_preservesFiniteProducts_and_equalizerCondition]
 
 end Profinite
@@ -88,7 +90,7 @@ end Profinite
 namespace Stonean
 
 theorem isSheaf_iff_preservesFiniteProducts
-    (F : Stonean.{u}ᵒᵖ ⥤ Type (u+1)) :
+    (F : Stonean.{u}ᵒᵖ ⥤ Type w) :
     IsSheaf (coherentTopology Stonean) F ↔ Nonempty (PreservesFiniteProducts F) := by
   rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left ?_]
   · rw [isSheaf_iff_isSheaf_of_type, extensiveTopology,
@@ -99,21 +101,24 @@ theorem isSheaf_iff_preservesFiniteProducts
     exact isSheafFor_regular_of_projective R F
 
 theorem isSheaf_iff_preservesFiniteProducts'
-    {A : Type (u+2)} [Category.{u+1} A] (G : A ⥤ Type (u+1))
-    [HasLimits A] [PreservesLimits G] [G.ReflectsIsomorphisms] (F : Stonean.{u}ᵒᵖ ⥤ A) :
+    {A : Type*} [Category A] (G : A ⥤ Type w)
+    [HasLimitsOfSize.{u, u+1} A] [PreservesLimitsOfSize.{u, u+1} G]
+    [G.ReflectsIsomorphisms] (F : Stonean.{u}ᵒᵖ ⥤ A) :
     Presheaf.IsSheaf (coherentTopology Stonean) F ↔
     Nonempty (PreservesFiniteProducts (F ⋙ G)) := by
-  rw [Presheaf.isSheaf_iff_isSheaf_forget (coherentTopology Stonean) F G,
+  rw [Presheaf.isSheaf_iff_isSheaf_comp (coherentTopology Stonean) F G,
     isSheaf_iff_preservesFiniteProducts]
 
 end Stonean
 
 namespace Condensed
 
-variable {A : Type (u+2)} [Category.{u+1} A] (G : A ⥤ Type (u+1)) [HasLimits A] [PreservesLimits G]
+variable {A : Type*} [Category A] (G : A ⥤ Type w)
+    [HasLimitsOfSize.{u, u+1} A] [PreservesLimitsOfSize.{u, u+1} G]
     [G.ReflectsIsomorphisms]
 
-instance : HasLimitsOfSize.{u, u+1} A := hasLimitsOfSizeShrink.{u, u+1, u+1, u+1} _
+noncomputable instance : PreservesLimitsOfSize.{0, 0} G :=
+  preservesLimitsOfSizeShrink.{0, u+1, 0, u} _
 
 /-- The condensed set associated to a finite-product-preserving presheaf on `Stonean`. -/
 noncomputable def ofSheafStonean (F : Stonean.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F] :
@@ -195,16 +200,19 @@ variable (R : Type (u+1)) [Ring R]
 /-- A `CondensedMod` version of `Condensed.ofSheafStonean`. -/
 noncomputable abbrev ofSheafStonean (F : Stonean.{u}ᵒᵖ ⥤ ModuleCat.{u+1} R)
     [PreservesFiniteProducts F] : CondensedMod R :=
+  haveI : HasLimitsOfSize.{u, u+1} (ModuleCat R) := hasLimitsOfSizeShrink.{u, u+1, u+1, u+1} _
   Condensed.ofSheafStonean (forget _) F
 
 /-- A `CondensedMod` version of `Condensed.ofSheafProfinite`. -/
 noncomputable abbrev ofSheafProfinite (F : Profinite.{u}ᵒᵖ ⥤ ModuleCat.{u+1} R)
     [PreservesFiniteProducts F] (hF : EqualizerCondition (F ⋙ forget _)) : CondensedMod R :=
+  haveI : HasLimitsOfSize.{u, u+1} (ModuleCat R) := hasLimitsOfSizeShrink.{u, u+1, u+1, u+1} _
   Condensed.ofSheafProfinite (forget _) F hF
 
 /-- A `CondensedMod` version of `Condensed.ofSheafCompHaus`. -/
 noncomputable abbrev ofSheafCompHaus (F : CompHaus.{u}ᵒᵖ ⥤ ModuleCat.{u+1} R)
     [PreservesFiniteProducts F] (hF : EqualizerCondition (F ⋙ forget _)) : CondensedMod R :=
+  haveI : HasLimitsOfSize.{u, u+1} (ModuleCat R) := hasLimitsOfSizeShrink.{u, u+1, u+1, u+1} _
   Condensed.ofSheafCompHaus (forget _) F hF
 
 end CondensedMod