oid al ▁category ▁is ▁a ▁category ▁ ▁equ ipped ▁with ▁a ▁mon oid al ▁structure . ▁A ▁mon oid al ▁structure ▁consists ▁of ▁the ▁following : ▁a ▁b if un ctor ▁ ▁called ▁the ▁tensor ▁product ▁or ▁mon oid al ▁product , ▁an ▁object ▁ ▁called ▁the ▁unit ▁object ▁or ▁identity ▁object , ▁three ▁natural ▁is omorph isms ▁subject ▁to ▁certain ▁coh er ence ▁conditions ▁express ing ▁the ▁fact ▁that ▁the ▁tensor ▁operation ▁is ▁associ ative : ▁there ▁is ▁a ▁natural ▁( in ▁each ▁of ▁three ▁arguments ▁, ▁, ▁) ▁isomorphism ▁, ▁called ▁associ ator , ▁with ▁components ▁, ▁has ▁ ▁as ▁left ▁and ▁right ▁identity : ▁there ▁are ▁two ▁natural ▁is omorph isms ▁ ▁and ▁, ▁respectively ▁called ▁left ▁and ▁right ▁un itor , ▁with ▁components ▁ ▁and ▁. ▁ ▁Note ▁that ▁a ▁good ▁way ▁to ▁remember ▁how ▁ ▁and ▁ ▁act ▁is ▁by ▁all iter ation ; ▁L ambda , ▁, ▁can c els ▁the ▁identity ▁on ▁the ▁left , ▁while ▁R ho , ▁, ▁can c els ▁the ▁identity ▁on ▁the ▁right . ▁ ▁The ▁coh er ence ▁conditions ▁for ▁these ▁natural ▁transformations ▁are : ▁▁ ▁for ▁all ▁, ▁, ▁ ▁and ▁ ▁in ▁, ▁the ▁pent agon ▁diagram ▁▁ ▁comm utes ; ▁ ▁for ▁all ▁ ▁and ▁ ▁in ▁, ▁the ▁triangle ▁diagram ▁▁ ▁comm utes ; ▁ ▁A ▁strict ▁mon oid al ▁category ▁is ▁one ▁for ▁which ▁the ▁natural ▁is omorph isms ▁α , ▁ λ ▁and ▁ ρ ▁are ▁ident ities . ▁Every ▁mon oid al ▁category