Returning to the case where M is the category of sets with cartesian product, the morphisms become functions from the one-point set I and must then, for any given object a, identify a particular element of each set , something we can then think of as the "identity morphism for a in C". Commutativity of the latter two diagrams is then the statement that compositions (as defined by the functions °) involving these distinguished individual "identity morphisms in C" behave exactly as per the identity rules for ordinary categories.
rEturninG to the cASE wHere m is the caTegOry of sets WITh cArtEsian produCt, ThE moRphIsms bECoMe functiOns frOM tHe one-poiNT SEt I and musT thEn, FOr Any Given ObjecT A, IdEnTiFY a paRticuLar eLEMeNt of each seT , SometHinG We cAn tHen thiNk of As ThE "idenTIty moRPhiSm for A in c". commuTativIty oF the lAtTeR two diagrams iS ThEn thE sTatemeNT That COMpOsiTionS (aS deFined bY The FUNCTioNs °) invOlvIng these dIstingUIshed INDiviDuAL "iDEntIty morphisms In C" BeHave exacTly as PeR the identIty rulEs for OrDINaRy CategorIes.