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.