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diff --git a/Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda b/Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda
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+++ b/Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda
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+{-# OPTIONS --without-K --safe #-}
+
+module Preorder.Primitive.MonotoneMap.Monoidal.Lax where
+
+open import Level using (Level; _⊔_)
+open import Preorder.Primitive.Monoidal using (MonoidalPreorder; SymmetricMonoidalPreorder)
+open import Preorder.Primitive.MonotoneMap using (MonotoneMap)
+
+record MonoidalMonotone
+    {c₁ c₂ ℓ₁ ℓ₂ : Level}
+    (P : MonoidalPreorder c₁ ℓ₁)
+    (Q : MonoidalPreorder c₂ ℓ₂)
+  : Set (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂) where
+
+  private
+    module P = MonoidalPreorder P
+    module Q = MonoidalPreorder Q
+
+  field
+    F : MonotoneMap P.U Q.U
+
+  open MonotoneMap F public
+
+  field
+    ε : Q.unit Q.≲ map P.unit
+    ⊗-homo : (p₁ p₂ : P.Carrier) → map p₁ Q.⊗ map p₂ Q.≲ map (p₁ P.⊗ p₂)
+
+record SymmetricMonoidalMonotone
+    {c₁ c₂ ℓ₁ ℓ₂ : Level}
+    (P : SymmetricMonoidalPreorder c₁ ℓ₁)
+    (Q : SymmetricMonoidalPreorder c₂ ℓ₂)
+  : Set (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂) where
+
+  private
+    module P = SymmetricMonoidalPreorder P
+    module Q = SymmetricMonoidalPreorder Q
+
+  field
+    monoidalMonotone : MonoidalMonotone P.monoidalPreorder Q.monoidalPreorder
+
+  open MonoidalMonotone monoidalMonotone public