This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

The precategory of functors and natural transformations from small to large precategories

module category-theory.precategory-of-functors-from-small-to-large-precategories where
Imports 
open import category-theory.functors-from-small-to-large-precategories
open import category-theory.large-precategories
open import category-theory.natural-transformations-functors-from-small-to-large-precategories
open import category-theory.precategories

open import foundation.identity-types
open import foundation.strictly-involutive-identity-types
open import foundation.universe-levels

Idea

Functors from a small precategory C to a large precategory D and natural transformations between them form a large precategory whose identity map and composition structure are inherited pointwise from the codomain precategory. This is called the precategory of functors from small to large precategories.

Definitions

The large precategory of functors and natural transformations from a small to a large precategory

module _
  {l1 l2 : Level} {α : Level → Level} {β : Level → Level → Level}
  (C : Precategory l1 l2)
  (D : Large-Precategory α β)
  where

  hom-functor-large-precategory-Small-Large-Precategory :
    {γF γG : Level}
    (F : functor-Small-Large-Precategory C D γF)
    (G : functor-Small-Large-Precategory C D γG) →
    UU (l1 ⊔ l2 ⊔ β γF γG)
  hom-functor-large-precategory-Small-Large-Precategory =
    natural-transformation-Small-Large-Precategory C D

  comp-hom-functor-large-precategory-Small-Large-Precategory :
    {γF γG γH : Level}
    {F : functor-Small-Large-Precategory C D γF}
    {G : functor-Small-Large-Precategory C D γG}
    {H : functor-Small-Large-Precategory C D γH} →
    natural-transformation-Small-Large-Precategory C D G H →
    natural-transformation-Small-Large-Precategory C D F G →
    natural-transformation-Small-Large-Precategory C D F H
  comp-hom-functor-large-precategory-Small-Large-Precategory {F = F} {G} {H} =
    comp-natural-transformation-Small-Large-Precategory C D F G H

  associative-comp-hom-functor-large-precategory-Small-Large-Precategory :
    {γF γG γH γI : Level}
    {F : functor-Small-Large-Precategory C D γF}
    {G : functor-Small-Large-Precategory C D γG}
    {H : functor-Small-Large-Precategory C D γH}
    {I : functor-Small-Large-Precategory C D γI}
    (h : natural-transformation-Small-Large-Precategory C D H I)
    (g : natural-transformation-Small-Large-Precategory C D G H)
    (f : natural-transformation-Small-Large-Precategory C D F G) →
    comp-natural-transformation-Small-Large-Precategory C D F G I
      ( comp-natural-transformation-Small-Large-Precategory C D G H I h g)
      ( f) ＝
    comp-natural-transformation-Small-Large-Precategory C D F H I
      ( h)
      ( comp-natural-transformation-Small-Large-Precategory C D F G H g f)
  associative-comp-hom-functor-large-precategory-Small-Large-Precategory
    {F = F} {G} {H} {I} h g f =
    associative-comp-natural-transformation-Small-Large-Precategory
      C D F G H I f g h

  involutive-eq-associative-comp-hom-functor-large-precategory-Small-Large-Precategory :
    {γF γG γH γI : Level}
    {F : functor-Small-Large-Precategory C D γF}
    {G : functor-Small-Large-Precategory C D γG}
    {H : functor-Small-Large-Precategory C D γH}
    {I : functor-Small-Large-Precategory C D γI}
    (h : natural-transformation-Small-Large-Precategory C D H I)
    (g : natural-transformation-Small-Large-Precategory C D G H)
    (f : natural-transformation-Small-Large-Precategory C D F G) →
    comp-natural-transformation-Small-Large-Precategory C D F G I
      ( comp-natural-transformation-Small-Large-Precategory C D G H I h g)
      ( f) ＝ⁱ
    comp-natural-transformation-Small-Large-Precategory C D F H I
      ( h)
      ( comp-natural-transformation-Small-Large-Precategory C D F G H g f)
  involutive-eq-associative-comp-hom-functor-large-precategory-Small-Large-Precategory
    {F = F} {G} {H} {I} h g f =
    involutive-eq-associative-comp-natural-transformation-Small-Large-Precategory
      C D F G H I f g h

  id-hom-functor-large-precategory-Small-Large-Precategory :
    {γF : Level} {F : functor-Small-Large-Precategory C D γF} →
    natural-transformation-Small-Large-Precategory C D F F
  id-hom-functor-large-precategory-Small-Large-Precategory {F = F} =
    id-natural-transformation-Small-Large-Precategory C D F

  left-unit-law-comp-hom-functor-large-precategory-Small-Large-Precategory :
    {γF γG : Level}
    {F : functor-Small-Large-Precategory C D γF}
    {G : functor-Small-Large-Precategory C D γG}
    (a : natural-transformation-Small-Large-Precategory C D F G) →
    comp-natural-transformation-Small-Large-Precategory C D F G G
      ( id-natural-transformation-Small-Large-Precategory C D G) a ＝
    a
  left-unit-law-comp-hom-functor-large-precategory-Small-Large-Precategory
    { F = F} {G} =
    left-unit-law-comp-natural-transformation-Small-Large-Precategory C D F G

  right-unit-law-comp-hom-functor-large-precategory-Small-Large-Precategory :
    {γF γG : Level}
    {F : functor-Small-Large-Precategory C D γF}
    {G : functor-Small-Large-Precategory C D γG}
    (a : natural-transformation-Small-Large-Precategory C D F G) →
    comp-natural-transformation-Small-Large-Precategory C D F F G
        a (id-natural-transformation-Small-Large-Precategory C D F) ＝
    a
  right-unit-law-comp-hom-functor-large-precategory-Small-Large-Precategory
    { F = F} {G} =
    right-unit-law-comp-natural-transformation-Small-Large-Precategory C D F G

  functor-large-precategory-Small-Large-Precategory :
    Large-Precategory (λ l → l1 ⊔ l2 ⊔ α l ⊔ β l l) (λ l l' → l1 ⊔ l2 ⊔ β l l')
  obj-Large-Precategory functor-large-precategory-Small-Large-Precategory =
    functor-Small-Large-Precategory C D
  hom-set-Large-Precategory functor-large-precategory-Small-Large-Precategory =
    natural-transformation-set-Small-Large-Precategory C D
  comp-hom-Large-Precategory
    functor-large-precategory-Small-Large-Precategory {X = F} {G} {H} =
    comp-hom-functor-large-precategory-Small-Large-Precategory {F = F} {G} {H}
  id-hom-Large-Precategory
    functor-large-precategory-Small-Large-Precategory {X = F} =
    id-hom-functor-large-precategory-Small-Large-Precategory {F = F}
  involutive-eq-associative-comp-hom-Large-Precategory
    functor-large-precategory-Small-Large-Precategory {X = F} {G} {H} {I} =
    involutive-eq-associative-comp-hom-functor-large-precategory-Small-Large-Precategory
      { F = F} {G} {H} {I}
  left-unit-law-comp-hom-Large-Precategory
    functor-large-precategory-Small-Large-Precategory {X = F} {G} =
    left-unit-law-comp-hom-functor-large-precategory-Small-Large-Precategory
      { F = F} {G}
  right-unit-law-comp-hom-Large-Precategory
    functor-large-precategory-Small-Large-Precategory {X = F} {G} =
    right-unit-law-comp-hom-functor-large-precategory-Small-Large-Precategory
      { F = F} {G}

The small precategories of functors and natural transformations from a small to a large precategory

module _
  {l1 l2 : Level} {α : Level → Level} {β : Level → Level → Level}
  (C : Precategory l1 l2)
  (D : Large-Precategory α β)
  where

  functor-precategory-Small-Large-Precategory :
    (l : Level) → Precategory (l1 ⊔ l2 ⊔ α l ⊔ β l l) (l1 ⊔ l2 ⊔ β l l)
  functor-precategory-Small-Large-Precategory =
    precategory-Large-Precategory
      ( functor-large-precategory-Small-Large-Precategory C D)