C ) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors . Every functor F  : D ? E induces a f...

Diagonal functor : The diagonal functor is defined as the functor from D to the functor category D C which sends each object in D to the constant f...

Yoneda's lemma asserts that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full...

Preservation of limits Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covari...

contravariant functor is right exact if and only if it turns finite limits into colimits. A functor is exact if and only if it is both left exact a...

The distinction is particularly important for computations with tensors , which often have mixed variance (both covariant and contravariant compone...

Functors Main article: functor Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of al...

If the diagram is contravariant then it is called an inverse system . [ edit ] Cones and limits A cone of a diagram D  : J ? C is a morphism from t...

Spec even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces . In fact it is the unive...

In practice, for a valid statement about a particular category , the dual statement is valid in the dual category ( ). [ edit ] Duality The example...

Transformation of local covariant basis in the case of general curvilinear coordinates As stated above, contravariant vectors are vectors with cont...