CategoryTheory.Idempotents.toKaroubi_comp_karoubiFunctorCategoryEmbedding

theorem CategoryTheory.Idempotents.toKaroubi_comp_karoubiFunctorCategoryEmbedding : ∀ (J : Type u_1) (C : Type u_2) [inst : CategoryTheory.Category.{u_3, u_1} J] [inst_1 : CategoryTheory.Category.{u_4, u_2} C], (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.Functor J C)).comp (CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding J C) = (CategoryTheory.whiskeringRight J C (CategoryTheory.Idempotents.Karoubi C)).obj (CategoryTheory.Idempotents.toKaroubi C)

The theorem `CategoryTheory.Idempotents.toKaroubi_comp_karoubiFunctorCategoryEmbedding` states that for any types `J` and `C` which are categories, the composition of the functor from `J` to `C` to the Karoubi envelope of `J` to `C` with the functor from the Karoubi envelope of `J` to `C` to `J` to the Karoubi envelope of `C` is equal to the functor from `J` to `C` to `J` to the Karoubi envelope of `C` given by the composition with the functor `toKaroubi C` from `C` to its Karoubi envelope. This highlights a key relationship between these functors and the transformation process to the Karoubi envelope.

More concisely: For any categories J and C, the functor from J to the Karoubi envelope of C, followed by the embedding of the Karoubi envelope of J in C, is equal to the composition of the embedding of J in the Karoubi envelope of C, the functor from J to C, and the embedding of C in its Karoubi envelope.