[Merged by Bors] - feat: monoidal category structure on coalgebras

[101damnations]

Monoidal category structure on coalgebras, using the definitions in Mathlib.RingTheory.Coalgebra.TensorProduct added in #11975

---

We make a MonoidalCategoryStruct instance using the definitions from #11975, and upgrade this to a MonoidalCategory instance using Monoidal.induced and the forgetful functor to modules. See #11974 and #11975 for more info on these PRs.

• depends on: [Merged by Bors] - feat: coalgebra homomorphisms #11961
• depends on: [Merged by Bors] - feat(Algebra/Module): Use coercion from SemilinearEquivClass to SemilinearEquiv #11966
• depends on: [Merged by Bors] - feat: coalgebra isomorphisms #11970
• depends on: [Merged by Bors] - feat: category of coalgebras #11972
• depends on: [Merged by Bors] - feat: category equivalence between coalgebras and comonoid objects #11974
• depends on: [Merged by Bors] - feat: tensor products of coalgebras #11975

[github-actions]

PR summary 81743901f1

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.Category.CoalgebraCat.Monoidal	1115

---

Declarations diff

+ MonoidalCategory.inducingFunctorData
+ instMonoidalCategory
+ instMonoidalCategoryStruct

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.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat: coalgebra homomorphisms #11961
• [Merged by Bors] - feat(Algebra/Module): Use coercion from SemilinearEquivClass to SemilinearEquiv #11966
• [Merged by Bors] - feat: coalgebra isomorphisms #11970
• [Merged by Bors] - feat: category of coalgebras #11972
• [Merged by Bors] - feat: category equivalence between coalgebras and comonoid objects #11974
• [Merged by Bors] - feat: tensor products of coalgebras #11975
  By Dependent Issues (🤖). Happy coding!

[joelriou]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• Build
• Lint style