feat(Algebra/Category): P.Under ⊤ R has finite limits if P is stable under finite products and equalizers

[chrisflav]

From Pi1.

---

• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): functoriality of coconeLeftOpOfCone and co #36918
• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): reflecting (co)limits and opposites #36909
• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): creation of limits and opposites #37504
• depends on: [Merged by Bors] - chore(CategoryTheory/Limits): dualize existence of limits in Over to Under #38012
• depends on: chore(CategoryTheory/Limits): limit properties of MorphismProperty.Under #38180

[github-actions]

PR summary a090f46da7

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
4 files Mathlib.Algebra.Category.CommAlgCat.FiniteType Mathlib.CategoryTheory.Comma.Over.StrictInitial Mathlib.CategoryTheory.MorphismProperty.Comma Mathlib.CategoryTheory.MorphismProperty.OverAdjunction	1
Mathlib.Algebra.Category.Ring.Under.Property (new file)	2664

---

Declarations diff

+ CreatesFiniteProducts.mk'
+ HasEqualizers
+ HasFiniteProducts
+ StructuredArrow.closedUnderColimitsOfShape_discrete_empty
+ StructuredArrow.isClosedUnderLimitsOfShape
+ Under.closedUnderColimitsOfShape_discrete_empty
+ Under.closedUnderColimitsOfShape_pushout
+ Under.createsEqualizersForget
+ Under.createsFiniteLimitsForget
+ Under.createsFiniteProductsForget
+ Under.hasEqualizers
+ Under.hasFiniteLimits
+ Under.hasFiniteProducts
+ Under.property_limit_of_hasFiniteProducts_of_hasEqualizers
+ createsLimitsOfShape_walkingCospan
+ instance (priority := 900) [HasPushouts T] [P.IsStableUnderComposition]
+ instance : CreatesFiniteColimits (Under.forget P ⊤ X)
+ instance : PreservesFiniteColimits (Under.forget P ⊤ X)
+ instance [HasFiniteWidePushouts T] : HasFiniteColimits (P.Under ⊤ X)
+ instance [P.ContainsIdentities] (Y : P.Under ⊤ X) :
+ instance [P.ContainsIdentities] : HasInitial (P.Under ⊤ X)
+ instance [P.ContainsIdentities] [P.RespectsIso] :
+ inverseImage_op_overObj
+ inverseImage_op_underObj
+ mkIdInitial
+ structuredArrow_iso_iff
+ under_iso_iff

You can run this locally as follows

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

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

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

---

Increase in tech debt: (relative, absolute) = (6.00, 0.00)

Current number	Change	Type
6519	6	backward.isDefEq.respectTransparency

Current commit cdd5d3e41d
Reference commit a090f46da7

You can run this locally as

./scripts/reporting/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).

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - chore(CategoryTheory/Limits): functoriality of coconeLeftOpOfCone and co #36918
• [Merged by Bors] - chore(CategoryTheory/Limits): reflecting (co)limits and opposites #36909
• [Merged by Bors] - chore(CategoryTheory/Limits): creation of limits and opposites #37504
• [Merged by Bors] - chore(CategoryTheory/Limits): dualize existence of limits in Over to Under #38012
• chore(CategoryTheory/Limits): limit properties of MorphismProperty.Under #38180
  By Dependent Issues (🤖). Happy coding!