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

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

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

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

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

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

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

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

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.