In other words, a functor F : C → D is full if the maps
FX,Y : MorC(X, Y) → MorD(FX, FY)
are surjective for every pair of objects X and Y in C.
Note that a full functor need not be surjective on objects or morphisms. That is, there may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C.
For example, let F : C → Set be the functor which maps every object in C to the empty set and every morphism to the empty function. Then F is full, but neither surjective on objects or morphisms.
Another example is the forgetful functorAb → Grp. This is full, but neither surjective on objects or morphisms. A counterexample is the forgetful functor Grp → Set. This is not full as there are functions between group which are not group homomorphisms.
Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
In English, an adjectival phrase may occur as a postmodifier to a noun (a bin full of toys), or as a predicate to a verb (the bin is full of toys).
In other languages, some sort of grammatical functor between the two nouns may be required.
These attributive nouns are not classed as adjectives, and they cannot be used in post-position; while the majority of adjectives can function both attributively and predicatively, an attributive noun cannot be made predicative by simply putting it after the head word.
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