Whats the motivation behind the definition of naturality?

We know that a natural transformation between functors is a collection of morphisms which satisfies a certain commutative diagram. I have my own ways to to understand this: Naturality vaguely means that no artificial/inconsistent system of choices have been made, which means that the collection of morphisms are not biased towards any particular thing, which means that the collection of morphisms must behave " nicely " with everything else. Now since the objects we are concerned with are a collection of morphisms, the term " everything else " just means " every morphism that can interact with the collection of morphisms " and this nice interaction ammounts to the naturality square being commutative. In certain cases like G-Sets or morphisms of sheaves it's relatively easy to understand the definition of natural transformations because here they are just structure preserving maps, and for the general case we just generalize these specific observations. In category theory/algebra one often defines a morphism to be something that commutes with everything else that already exists, for example morphisms of chain complexes, so the same goes for natural transformations. What were Eilenberg and Mac Lane thinking while defining naturality? Do you guys have any nice motivation?