proposition: Lemma lawvere_eqv_fmap_counit_unit {a} : fmap[G] lawvere_eqv_counit ∘ lawvere_eqv_unit ≈ id[G a].
proof: Proof. simpl; intros. unfold lawvere_eqv_counit, lawvere_eqv_unit. rewrite <- (id_right (to lawvere_morph_iso _)). rewrite comp_assoc. rewrite <- lawvere_to_morph_iso_functorial. simpl (`2 _). rewrite fmap_id, !id_right. now sapply (iso_to_from (@lawvere_morph_iso (G a) a)). Qed.
imports: Category.Lib Category.Theory.Category Category.Theory.Isomorphism Category.Theory.Functor Category.Theory.Natural.Transformation Category.Theory.Adjunction Category.Adjunction.Natural.Transformation Category.Adjunction.Natural.Transformation.Universal Category.Construction.Comma Category.Construction.Product Category.Instance.Cat Category.Instance.Fun Category.Instance.Sets