If 1945 be the date for the birth of categories then 1942 is the date for their conception: the first and last paragraphs of the Eilenberg and Mac Lane paper "Natural isomorphisms in group theory" Proc. Nat. Acad. Sci. USA. 28, (1942). 537--543. (Communicated October 26, 1942): Frequently in modern mathematics there occur phenomena of "naturality": a "natural" isomorphism between two groups or between two complexes, a "natural" homeomorphism of two spaces and the like. We here propose a precise definition of the "naturality" of such correspondences, as a basis for an appropriate general theory. In this preliminary report we restrict ourselves to the natural isomorphism of group theory; with this limitation we can present the basic concepts of our theory without developing the axiomatic approach necessary for a general treatment applicable to various branches of mathematics. * * * An inspection of the concept of a functor and of a natural equivalence shows that they may be applied not only to groups with their homomorphisms, but also to topological spaces with their continuous mappings, to simplicial complexes with their simplicial transformations, and to Banach spaces with their linear transformations. These and similar applications can be embodied in a suitable axiomatic theory. The resulting much wider concept of naturality, as an equivalence between functors, will be studied in a subsequent paper.