A category consists of a collection of objects, along with a collection of morphisms (also called arrows) between these objects. Morphisms represent relationships or transformations between objects.

A morphism is a structure-preserving map between two objects in a category. It represents a relationship or transformation between these objects that preserves the structure of the category.

Composition of morphisms is a fundamental operation in category theory. It allows us to combine two morphisms, one after the other, to obtain a new morphism. The composition of morphisms is associative, meaning that the order in which they are composed does not matter.

A functor is a structure-preserving map between two categories. It assigns objects and morphisms of one category to objects and morphisms of another category in a way that preserves the structure of the categories.

A category consists of a collection of objects and a collection of morphisms between these objects. A set, on the other hand, is a collection of elements without any structure or relationships between them.

The category of sets and functions, the category of groups and homomorphisms, and the category of topological spaces and continuous maps are all examples of categories. These categories are fundamental in various branches of mathematics.

The forgetful functor from the category of groups to the category of sets, the functor that assigns to each vector space its dual space, and the functor that assigns to each topological space its fundamental group are all examples of functors. These functors are used in various mathematical constructions and applications.

The Yoneda lemma is a fundamental result in category theory that relates functors to natural transformations. It states that every functor from a category C to the category of sets is uniquely determined by the natural transformation it induces between the representable functor and the identity functor on C.

A natural transformation between two functors is a morphism that preserves the structure of the functors. It is a collection of morphisms, one for each object in the category, that commutes with the compositions of the functors.

An adjoint pair of functors consists of two functors, one called the left adjoint and the other called the right adjoint, such that there is a natural bijection between the hom-sets of the two functors.

A limit of a diagram in a category is an object that represents the universal property of the diagram. It is an object that satisfies a certain universal property with respect to all the objects in the diagram.

A colimit of a diagram in a category is an object that represents the universal property of the diagram. It is an object that satisfies a certain universal property with respect to all the objects in the diagram.

A limit is a universal object for a diagram, meaning that it satisfies a certain universal property with respect to all the objects in the diagram. A colimit is a universal object for a codiagram, which is the dual notion of a diagram.

The product of a family of objects, the equalizer of a pair of morphisms, and the kernel of a morphism are all examples of limits in category theory. These limits are used to construct new objects from existing objects in a category.

The coproduct of a family of objects, the coequalizer of a pair of morphisms, and the cokernel of a morphism are all examples of colimits in category theory. These colimits are used to construct new objects from existing objects in a category.