reserve B,C,D,C9,D9 for Category;
reserve E for Subcategory of C;

theorem
  the carrier of [:C,D:] = [:the carrier of C,the carrier of D:] & the
  carrier' of [:C,D:] = [:the carrier' of C,the carrier' of D:] & the Source of
  [:C,D:] = [:the Source of C,the Source of D:] & the Target of [:C,D:] = [:the
Target of C,the Target of D:] & the Comp of [:C,D:] = |:the Comp of C, the Comp
  of D:|;
