∗ . ▁ ▁A ▁category ▁C ▁en rich ed ▁in ▁a ▁mon oid al ▁category ▁M ▁rep laces ▁the ▁notion ▁of ▁a ▁set ▁of ▁morph isms ▁between ▁pairs ▁of ▁objects ▁in ▁C ▁with ▁the ▁notion ▁of ▁an ▁M - object ▁of ▁morph isms ▁between ▁every ▁two ▁objects ▁in ▁C . ▁ ▁Free ▁strict ▁mon oid al ▁category ▁▁ ▁For ▁every ▁category ▁C , ▁the ▁free ▁strict ▁mon oid al ▁category ▁ Σ ( C ) ▁can ▁be ▁constructed ▁as ▁follows : ▁ ▁its ▁objects ▁are ▁lists ▁( finite ▁sequences ) ▁A 1 , ▁... , ▁An ▁of ▁objects ▁of ▁C ; ▁ ▁there ▁are ▁ar rows ▁between ▁two ▁objects ▁A 1 , ▁... , ▁Am ▁and ▁B 1 , ▁... , ▁B n ▁only ▁if ▁m ▁= ▁n , ▁and ▁then ▁the ▁ar rows ▁are ▁lists ▁( finite ▁sequences ) ▁of ▁ar rows ▁f 1 : ▁A 1 ▁→ ▁B 1 , ▁... , ▁fn : ▁An ▁→ ▁B n ▁of ▁C ; ▁ ▁the ▁tensor ▁product ▁of ▁two ▁objects ▁ ▁A 1 , ▁... , ▁An ▁and ▁B 1 , ▁... , ▁B m ▁is ▁the ▁concaten ation ▁A 1 , ▁... , ▁An , ▁B 1 , ▁... , ▁B m ▁of ▁the ▁two ▁lists , ▁and , ▁similarly , ▁the ▁tensor ▁product ▁of ▁two ▁morph isms ▁is ▁given ▁by ▁the ▁concaten ation ▁of ▁lists . ▁The ▁identity ▁object ▁is ▁the ▁empty ▁list . ▁This ▁operation ▁ Σ ▁mapping ▁category ▁C ▁to ▁ Σ ( C ) ▁can ▁be ▁extended ▁to