[Merged by Bors] - feat(CategoryTheory/Enriched): functor categories are enriched #18009
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Let `C` be a category that is enriched over a monoidal category `V` in such a way that the category structure and the enriched category structure are compatible. Then, if `J` is a category and that `V` has certain limits, then the functor category `J ⥤ C` is also enriched over `V`. (Plan: using #17326, we may use this for `C := C` closed monoidal in order to show that a category of functors `J ⥤ C` to a monoidal category is enriched over `C`, and, by applying this to all `Under X` categories for `X : C`, it should follow that `J ⥤ C` is also closed monoidal. This should give a more explicit approach as compared to #16067.) Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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Let `C` be a category that is enriched over a monoidal category `V` in such a way that the category structure and the enriched category structure are compatible. Then, if `J` is a category and that `V` has certain limits, then the functor category `J ⥤ C` is also enriched over `V`. (Plan: using #17326, we may use this for `C := C` closed monoidal in order to show that a category of functors `J ⥤ C` to a monoidal category is enriched over `C`, and, by applying this to all `Under X` categories for `X : C`, it should follow that `J ⥤ C` is also closed monoidal. This should give a more explicit approach as compared to #16067.) Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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…ctor categories (#18414) Let `C` be a `V`-enriched ordinary category. Functor categories `J ⥤ C` have been `V`-enriched in #18009. Given two functors `F₁` and `F₂` in `J ⥤ C`, we use the previous results for functors `Under j ⥤ C` for all `j : J` in order to construct `functorEnrichedHom V F₁ F₂ : J ⥤ V`, and show that the limit of this functor identifies to `enrichedHom V F₁ F₂`. Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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Let
Cbe a category that is enriched over a monoidal categoryVin such a way that the category structure and the enriched category structure are compatible. Then, ifJis a category and thatVhas certain limits, then the functor categoryJ ⥤ Cis also enriched overV.(Plan: using #17326, we may use this for
C := Cclosed monoidal in order to show that a category of functorsJ ⥤ Cto a monoidal category is enriched overC, and, by applying this to allUnder Xcategories forX : C, it should follow thatJ ⥤ Cis also closed monoidal. This should give a more explicit approach as compared to #16067.)