obj_char: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and a :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "horizontal_homs.obj (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) (strictified_bicategory.src V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B) a = (partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B) a a) \<and> bicategorical_language.Obj V\<^sub>B src\<^sub>B (concrete_category.Dom a) \<and> concrete_category.Map a = Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Dom a))"