the identity to the dual (each space has an isomorphism to its dual, and the maps in the category are required to commute). Viewed in this light, this construction (add transforms for each object, restrict maps to commute with these) is completely general, and does not depend on any particular properties of vector spaces. In this category (finite-dimensional vector spaces with a nondegenerate bilinear form, maps linear transforms that respect the bilinear form), the dual of a map between vector spaces can be identified as a transpose. Often for reasons of geometric interest this is specialized to a subcategory, by requiring that the nondegenerate bilinear forms have additional properties, such as being symmetric (orthogonal matrices), symmetric and positive definite (inner product space), symmetric sesquilinear (Hermitian spaces), skew-symmetric and totally isotropic (symplectic vector space), etc. – in all these categories a vector space is naturally identified with its dual, by the nondegenerate bilinear form. Operations with natural transformations Vertical composition If and are natural transformations between functors , then we can compose them to get a natural transformation . This is done componentwise: . This vertical composition of natural transformations is associative and has an identity, and allows one to consider the collection of all functors itself as a category (see below under Functor categories). The identity natural transformation on functor has components . For , . Horizontal composition If is a natural transformation between functors and is a natural transformation between functors , then the composition of functors allows a composition of natural transformations with components . By using whiskering (see below), we can write , hence . This horizontal composition of natural transformations is also associative with identity. This identity is the identity natural transformation on the identity functor, i.e., the natural transformation that associate to each object its identity morphism: for object in category , . For with , . As identity functors and are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa. Whiskering Whiskering is an external binary operation between a functor and a natural transformation. If is a natural transformation between functors , and is another functor, then we can form the natural transformation by defining . If on the other hand is a functor, the natural transformation is defined by . It's also an horizontal composition where one of the natural transformations is the identity natural