The (Functor-)Category of Pre-Elgot Monads.

module Category.Construction.PreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient
open import Monad.PreElgot ambient
open import Algebra.Elgot cocartesian
open HomReasoning
open Equiv
open M C
open MR C

First we look at morphisms between pre-Elgot monads:

module _ (P S : PreElgotMonad) where
  private
    open PreElgotMonad P using () renaming (T to TP; elgotalgebras to P-elgots)
    open PreElgotMonad S using () renaming (T to TS; elgotalgebras to S-elgots)
    module TP = Monad TP
    module TS = Monad TS
    open RMonad (Monad⇒Kleisli C TP) using () renaming (extend to extendP)
    open RMonad (Monad⇒Kleisli C TS) using () renaming (extend to extendS)
    _#P = λ {X} {A} f → P-elgots._# {X} {A} f
    _#S = λ {X} {A} f → S-elgots._# {X} {A} f
  record PreElgotMonad-Morphism : Set (o ⊔ ℓ ⊔ e) where
    field
      α : NaturalTransformation TP.F TS.F
    module α = NaturalTransformation α
    field
      α-η : ∀ {X}
        → α.η X ∘ TP.η.η X ≈ TS.η.η X
      α-μ : ∀ {X}
        → α.η X ∘ TP.μ.η X ≈ TS.μ.η X ∘ TS.F.₁ (α.η X) ∘ α.η (TP.F.₀ X)
      preserves : ∀ {X A} (f : X ⇒ TP.F.₀ A + X) → α.η A ∘ f #P ≈ ((α.η A +₁ idC) ∘ f) #S

Now the category:

PreElgotMonads : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) (o ⊔ e)
PreElgotMonads = record
  { Obj = PreElgotMonad
  ; _⇒_ = PreElgotMonad-Morphism
  ; _≈_ = λ f g → (PreElgotMonad-Morphism.α f) ≃ (PreElgotMonad-Morphism.α g)
  ; id = id'
  ; _∘_ = _∘'_
  ; assoc = assoc
  ; sym-assoc = sym-assoc
  ; identityˡ = identityˡ
  ; identityʳ = identityʳ
  ; identity² = identity²
  ; equiv = record { refl = refl ; sym = λ f → sym f ; trans = λ f g → trans f g }
  ; ∘-resp-≈ = λ f≈h g≈i → ∘-resp-≈ f≈h g≈i
  }
  where
    open Elgot-Algebra-on using (#-resp-≈)
    id' : ∀ {A : PreElgotMonad} → PreElgotMonad-Morphism A A
    id' {A} = record
      { α = ntHelper (record
        { η = λ _ → idC
        ; commute = λ _ → id-comm-sym
        })
      ; α-η = identityˡ
      ; α-μ = sym (begin
        T.μ.η _ ∘ T.F.₁ idC ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
        T.μ.η _ ∘ T.F.₁ idC       ≈⟨ elimʳ T.F.identity ⟩
        T.μ.η _                   ≈⟨ sym identityˡ ⟩
        idC ∘ T.μ.η _             ∎)
      ; preserves = λ f → begin
        idC ∘ f #            ≈⟨ identityˡ ⟩
        f #                  ≈⟨ sym (#-resp-≈ elgotalgebras (elimˡ ([]-unique id-comm-sym id-comm-sym))) ⟩
        ((idC +₁ idC) ∘ f) # ∎
      }
      where
        open PreElgotMonad A using (T; elgotalgebras)
        module T = Monad T
        _# = λ {X} {A} f → elgotalgebras._# {X} {A} f
    _∘'_ : ∀ {X Y Z : PreElgotMonad} → PreElgotMonad-Morphism Y Z → PreElgotMonad-Morphism X Y → PreElgotMonad-Morphism X Z
    _∘'_ {X} {Y} {Z} f g = record
      { α = αf ∘ᵥ αg
      ; α-η = λ {A} → begin
        (αf.η A ∘ αg.η A) ∘ TX.η.η A ≈⟨ pullʳ (α-η g) ⟩
        αf.η A ∘ TY.η.η A            ≈⟨ α-η f ⟩
        TZ.η.η A                     ∎
      ; α-μ = λ {A} → begin
        (αf.η A ∘ αg.η A) ∘ TX.μ.η A                                                       ≈⟨ pullʳ (α-μ g) ⟩
        αf.η A ∘ TY.μ.η A ∘ TY.F.₁ (αg.η A) ∘ αg.η (TX.F.₀ A)                              ≈⟨ pullˡ (α-μ f) ⟩
        (TZ.μ.η A ∘ TZ.F.₁ (αf.η A) ∘ αf.η (TY.F.₀ A)) ∘ TY.F.₁ (αg.η A) ∘ αg.η (TX.F.₀ A) ≈⟨ assoc ○ refl⟩∘⟨ pullʳ (pullˡ (NaturalTransformation.commute αf (αg.η A))) ⟩
        TZ.μ.η A ∘ TZ.F.₁ (αf.η A) ∘ (TZ.F.₁ (αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A) ≈⟨ refl⟩∘⟨ pullˡ (pullˡ (sym (Functor.homomorphism TZ.F))) ⟩
        TZ.μ.η A ∘ (TZ.F.₁ (αf.η A ∘ αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A)          ≈⟨ refl⟩∘⟨ assoc ⟩
        TZ.μ.η A ∘ TZ.F.₁ ((αf.η A ∘ αg.η A)) ∘  αf.η (TX.F.₀ A) ∘ αg.η (TX.F.₀ A)         ∎
      ; preserves = λ {A} {B} h → begin
        (αf.η B ∘ αg.η B) ∘ (h #X)                   ≈⟨ pullʳ (preserves g h) ⟩
        αf.η B ∘ ((αg.η B +₁ idC) ∘ h) #Y            ≈⟨ preserves f ((αg.η B +₁ idC) ∘ h) ⟩
        (((αf.η B +₁ idC) ∘ (αg.η B +₁ idC) ∘ h) #Z) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras Z) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
        (((αf.η B ∘ αg.η B +₁ idC) ∘ h) #Z)          ∎
      }
      where
        module TX = Monad (PreElgotMonad.T X)
        module TY = Monad (PreElgotMonad.T Y)
        module TZ = Monad (PreElgotMonad.T Z)
        _#X = λ {A} {B} f → PreElgotMonad.elgotalgebras._# X {A} {B} f
        _#Y = λ {A} {B} f → PreElgotMonad.elgotalgebras._# Y {A} {B} f
        _#Z = λ {A} {B} f → PreElgotMonad.elgotalgebras._# Z {A} {B} f

        open PreElgotMonad-Morphism using (α-η; α-μ; preserves)

        open PreElgotMonad-Morphism f using () renaming (α to αf)
        open PreElgotMonad-Morphism g using () renaming (α to αg)