feat(CategoryTheory/Limits/Shapes/Pullback/Categorical): pseudofunctoriality structure of categorical pullback squares

[robin-carlier]

Using pseudofunctoriality of the construction CatCommSqOver, we show that categorical pullback squares have pseudofunctorial-like structures: given a CatCommSq T L R B and a CatPullbackSq T' L' R' B', and a CatCospanTransform R B R' B', there is a
functor CatPullbackSq.functorOfTransform between the top left corners of the two squares. We give CatCommSq instances that relates this functor with other constructions such as CatCommSqOver.transform.
Similarly, we construct CatPullbackSq.functorOfTransform₂, that maps morphisms of CatCospanTransforms to natural transformations. Finally, we show that functorOfTransform respects identities and compositions.

We do not yet prove that all of this data satisfies all the expected properties of pseudofunctors: this will be the content of a follow-up PR.

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• depends on: feat(CategoryTheory/Limits/Shapes/Pullback/Categorical): Categorical pullback squares #27214

[github-actions]

PR summary 9c18b42ca8

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic	1
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Square (new file)	320

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Declarations diff

+ CatPullbackSquare
+ IsCatPullbackSquare
+ Iso.hom_inv_id_app_fst_app
+ Iso.hom_inv_id_app_snd_app
+ Iso.hom_inv_id_fst_app
+ Iso.hom_inv_id_snd_app
+ Iso.inv_hom_id_app_fst_app
+ Iso.inv_hom_id_app_snd_app
+ Iso.inv_hom_id_fst_app
+ Iso.inv_hom_id_snd_app
+ catPullbackSquare
+ counitIsoAppFst
+ counitIsoAppSnd
+ equivalence
+ functor
+ functorEquiv
+ functorEquivFunctorIdIso
+ functorEquivFunctorWhiskeringFunctorOfTransformObjSquare
+ functorEquivInverseOfSquareIso
+ functorEquivInverseTransformObjSquare
+ functorEquivInverseWhiskeringIsoFst
+ functorEquivInverseWhiskeringIsoSnd
+ functorEquivInverse_map_app_fst
+ functorEquivInverse_map_app_snd
+ functorEquiv_counitInv_functor_comp_fst_app
+ functorEquiv_counitInv_functor_comp_snd_app
+ functorEquiv_functor_UnitIso_comp_fst_app
+ functorEquiv_functor_UnitIso_comp_snd_app
+ functorOfTransform
+ functorOfTransformObjComp
+ functorOfTransformObjComp_hom_app_fst
+ functorOfTransformObjComp_hom_app_snd
+ functorOfTransformObjComp_inv_app_fst
+ functorOfTransformObjComp_inv_app_snd
+ functorOfTransformObjFstSquare
+ functorOfTransformObjId
+ functorOfTransformObjSndSquare
+ functorOfTransform_map_app_fst
+ functorOfTransform_map_app_snd
+ functorOfTransform_obj_map_fst
+ functorOfTransform_obj_map_snd
+ instance (F : C₁ ⥤ C₂) (G : C₃ ⥤ C₂) :
+ instance :
+ instance : (inverse T L R B).IsEquivalence
+ instance : Inhabited (CatCommSqOver F G <| F ⊡ G)
+ inverse
+ isCatPullbackSquare_iff_isEquivalence_toFunctorToCategoricalPullback
+ ofSquare
+ precompose
+ precomposeEquivalenceInverseIsoDefault
+ precomposeObjComp
+ precomposeObjId
+ precomposeObjTransformObjSquare
+ precomposeObjTransformObjSquare_iso_hom_comp
+ precomposeObjTransformObjSquare_iso_hom_id
+ precomposeObjTransformObjSquare_iso_hom_naturality₂
+ precomposeToFunctorToCategoricalPullbackSquare
+ precompose_map_associator
+ precompose_map_leftUnitor
+ precompose_map_rightUnitor
+ precompose_map_whiskerLeft
+ precompose_map_whiskerRight
+ transform
+ transformObjComp
+ transformObjId
+ transformObjPrecomposeObjSquare
+ transformObjPrecomposeObjSquare_iso_hom_id
+ transformObjPrecomposeObjSquare_iso_hom_naturality₂
+ transformPrecomposeObjSquare_iso_hom_comp
+ transform_map_associator
+ transform_map_leftUnitor
+ transform_map_rightUnitor
+ transform_map_whiskerLeft
+ transform_map_whiskerRight
+ unitIso
+ whiskeringLeftFunctorEquivFunctorSquare

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• feat(CategoryTheory/Limits/Shapes/Pullback/Categorical): Categorical pullback squares #27214
  By Dependent Issues (🤖). Happy coding!