functorial

Functorial refers to the property of a mathematical construction that can be carried out in a way compatible with maps between objects, so that the construction becomes a functor between categories. In category theory, a functor assigns to every object a new object and to every morphism a new morphism, while preserving identities and composition. When a construction can be organized in this way, it is said to be functorial.

Formally, a functor F from a category C to a category D assigns to each object X in C an object F(X) in D, and to each morphism f: X → Y in C a morphism F(f): F(X) → F(Y) in D, such that F(id_X) = id_{F(X)} and F(g ∘ f) = F(g) ∘ F(f) for all composable f and g. If the direction of morphisms is reversed, the construction is described as contravariant and yields a contravariant functor. Functoriality thus encodes a precise sense in which a construction respects the structure of the input category.

Common examples include the power set construction P: Set → Set, which maps a function f: X → Y to P(f): P(X) → P(Y) sending a subset A ⊆ X to its image f[A]; this is covariant. Homology, cohomology, and various algebraic constructions yield functors from categories such as topological spaces or groups to abelian groups or other categories, expressing how these invariants behave under continuous maps or homomorphisms. In practice, establishing functoriality ensures that the construction interacts coherently with maps and compositional processes across mathematical contexts.