0 ▁is ▁the ▁mon oid al ▁unit . ▁ ▁Properties ▁and ▁associated ▁not ions ▁ ▁It ▁follows ▁from ▁the ▁three ▁defining ▁coh er ence ▁conditions ▁that ▁a ▁large ▁class ▁of ▁diag rams ▁( i . e . ▁diag rams ▁whose ▁morph isms ▁are ▁built ▁using ▁, ▁, ▁, ▁ident ities ▁and ▁tensor ▁product ) ▁comm ute : ▁this ▁is ▁Mac ▁Lane ' s ▁" co her ence ▁theorem ". ▁It ▁is ▁sometimes ▁in ac cur ately ▁stated ▁that ▁all ▁such ▁diag rams ▁comm ute . ▁ ▁There ▁is ▁a ▁general ▁notion ▁of ▁mon oid ▁object ▁in ▁a ▁mon oid al ▁category , ▁which ▁general izes ▁the ▁ordinary ▁notion ▁of ▁mon oid ▁ ▁from ▁abstract ▁algebra . ▁Ord inary ▁mon oid s ▁are ▁precisely ▁the ▁mon oid ▁objects ▁in ▁the ▁cart esian ▁mon oid al ▁category ▁Set . ▁Further , ▁any ▁strict ▁mon oid al ▁category ▁can ▁be ▁seen ▁as ▁a ▁mon oid ▁object ▁in ▁the ▁category ▁of ▁categories ▁Cat ▁( equ ipped ▁with ▁the ▁mon oid al ▁structure ▁induced ▁by ▁the ▁cart esian ▁product ). ▁ ▁Mon oid al ▁fun ctors ▁are ▁the ▁fun ctors ▁between ▁mon oid al ▁categories ▁that ▁preserve ▁the ▁tensor ▁product ▁and ▁mon oid al ▁natural ▁transformations ▁are ▁the ▁natural ▁transformations , ▁between ▁those ▁fun ctors , ▁which ▁are ▁" compatible " ▁with ▁the ▁tensor ▁product . ▁ ▁Every ▁mon oid al ▁category ▁can ▁be ▁seen ▁as ▁the ▁category ▁B ( ∗ , ▁ ∗ ) ▁of ▁a ▁b ic ategory ▁B ▁with ▁only ▁one ▁object , ▁denoted ▁