##  [Functor](/functor-2) 

 Definition

A mapping between categories that assigns to every object of a source category an object of a target category and to every morphism in the source a morphism in the target, preserving identity morphisms and composition.

 

 

 

 

 

 





## Principle

Principle

Functoriality requires two preservation laws: identities map to identities, and composition is respected (F(g ∘ f) = F(g) ∘ F(f)); these laws make functors the structure-preserving maps of category theory.

 

 

 

 

 





## Demonstration

Demonstration

The free abelian group construction F:Set→Ab sending a set S to the free abelian group Z[S] and a function f:S→T to the group homomorphism Z[f]:Z[S]→Z[T] given by linear extension is a functor: it preserves identities and composition by construction.

 

 

 

 

## Misapplication

Misapplication

Defining an assignment on objects without giving a rule for morphisms that respects composition, or claiming a set-theoretic function between underlying object-collections is a functor when it fails to take arrows to arrows compatibly.

 

 

 

 

 





## Consequence

Consequence

Correctly used, functors transport structure and universal properties between categories, allow comparison of categorical invariants, and provide the carriers for constructions such as limits, adjoints, and derived functors.

 

 

 

 

## Reversal

Reversal

A contravariant functor reverses arrow directions: it assigns to each morphism f:X→Y a morphism F(f):F(Y)→F(X) and satisfies F(g ∘ f) = F(f) ∘ F(g), contrasting covariant functors which preserve arrow direction.

 

 

 

 

 





## Boundary

Boundary

Applies to functors between categories (with well-defined objects and hom-sets/collections); pseudofunctors, profunctors, or mere object-level maps lie outside the strict functorial notion unless additional coherence data is provided.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often confused with ordinary functions between sets: a functor must act on both objects and morphisms and preserve categorical composition and identities, a stronger requirement than a mere mapping of objects.

 

 

 

 

 





## Synthesis

Synthesis

A functor is a structure-preserving map between categories that assigns objects and morphisms coherently so that identities and compositions are respected, enabling the transfer and comparison of categorical constructions.