Enriched category

In this case, each path leading to C(a, d) in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms a → b → c → d, i.e. elements from C(a, b), C(b, c) and C(c, d).

Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.

The usual functorial axioms are replaced with corresponding commutative diagrams involving these morphisms.

In detail, one has that the diagram commutes, which amounts to the equation where I is the unit object of M. This is analogous to the rule F(ida) = idF(a) for ordinary functors.

Additionally, one demands that the diagram commute, which is analogous to the rule F(fg)=F(f)F(g) for ordinary functors.