 ##  [Natural Transformation](/natural-transformation) 

  ##  [Natural Transformation](https://natural.quantumdictionary.io/natural-transformation-0) 

  
 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)


**Natural &amp; Formal Sciences Dictionary**

 
Definition

A family of morphisms (called components) η_X : F(X) → G(X) indexed by objects X of a source category, connecting two functors F,G : C → D, such that for every morphism f : X → Y in C the square G(f) ∘ η_X = η_Y ∘ F(f) commutes. It expresses a coherent, objectwise map between functors.

 
 ##  [Natural Transformation](https://mathlogic.quantumdictionary.io/natural-transformation-1) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A family of morphisms between two functors with the same domain and codomain, indexed by the objects of the domain category, such that for every morphism in the domain a naturality square commutes; it provides a canonical, structure-preserving comparison between functorial interpretations.

 
 ##  [Natural Transformation](https://puremath.quantumdictionary.io/natural-transformation-2) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

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.