A family of morphisms between two functors F and G from the same source category C to the same target category D: for each object X of C a component η_X : F(X) → G(X) so that for every morphism f:X→Y the square G(f) ∘ η_X = η_Y ∘ F(f) (the naturality condition) commutes.

Principle

Naturality demands coherence: the component maps must intertwine the actions of the functors on morphisms, making the family a canonical transformation of functors rather than an arbitrary collection of objectwise maps.

Demonstration

Consider the canonical map η_V:V → V** for finite-dimensional vector spaces V over a field k, where V** is the double dual. The family {η_V} is a natural transformation from the identity functor on finite-dimensional Vect_k to the double-dual functor because dualization commutes with linear maps and the naturality squares commute.

Misapplication

Declaring a collection of objectwise maps a natural transformation without checking commutativity of naturality squares, or assuming any componentwise isomorphism yields a natural isomorphism between functors.

Consequence

When valid, a natural transformation provides a canonical way to compare functors, yields the morphisms needed for defining adjoints, limits, colimits, and 2-categorical structures, and identifies when two functors are isomorphic in the functor category.

Reversal

A natural isomorphism is the inversion: a natural transformation whose every component is an isomorphism, which exhibits the two functors as the same up to coherent isomorphism; absence of naturality reduces the family to mere componentwise maps.

Boundary

Requires both functors to share the same source and target categories and well-defined components at every object; families defined only on a subclass of objects or between functors with different targets are not natural transformations in the strict sense.

Semantic Tension

Differs from an arbitrary 'family of morphisms' by the naturality requirement; also distinct from a mere pointwise isomorphism since naturality enforces compatibility with all morphisms in the source category.

Synthesis

A natural transformation is a coherent family of component morphisms between two functors, satisfying naturality squares that make the comparison canonical and functorial across the whole source category.