to a particular isomorphism, this requires showing that for any isomorphism , there is some with which it does not commute; in some cases a single automorphism works for all candidate isomorphisms while in other cases one must show how to construct a different for each isomorphism. The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance. This is similar (but more categorical) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is "not natural", or rather "not unique", as automorphisms exist that do not preserve the direct sum decomposition – see for example. Some authors distinguish notationally, using for a natural isomorphism and for an unnatural isomorphism, reserving for equality (usually equality of maps). Example: fundamental group of torus As an example of the distinction between the functorial statement and individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus. The homotopy groups of a product space are naturally the product of the homotopy groups of the components, with the isomorphism given by projection onto the two factors, fundamentally because maps into a product space are exactly products of maps into the components – this is a functorial statement. However, the torus (which is abstractly a product of two circles) has fundamental group isomorphic to , but the splitting is not natural. Note the use of , , and : This abstract isomorphism with a product is not natural, as some isomorphisms of do not preserve the product: the self-homeomorphism of (thought of as the quotient space ) given by (geometrically a Dehn twist about one of the generating curves) acts as this matrix on (it's in the general linear group of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product – equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components". Naturality is a categorical notion, and requires being very precise about exactly what data is given – the torus as a