## Category

A category is a collection of objects that are linked by arrows.

## Functor

A functor is a map between categories.

#### Definition

Let C and D be categories. A functor F from C to D is a mapping that:

• associates to each object $$X$$ in C an object $$F(X)$$ in D,
• associates to each morphism $$f \colon X \to Y$$ in C a morphism $$F(f) \colon F(X) \to F(Y)$$ in D such that the following two conditions hold:
• $$F(id_{x}) = id_{F(x)}$$ for every object $$X$$ in C,
• $$F(g \circ f) = F(g) \circ F(x)$$ for all morphisms $$f \colon X \to Y$$ and $$g \colon Y \to Z$$ in C.

## Natural Transformations

A natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved.

#### Definition

If F and G are functors between the categories C and D, then a natural transformation $$\eta$$ from F to G is a family of morphisms that satisfies two requirements.

• The natural transformation must associate to every object $$X$$ in C a morphism $$\eta_{X} \colon F(X) \to G(X)$$ between objects of D. The morphism $$\eta_{X}$$ is called the component of $$\eta$$ at $$X$$.
• Components must be such that for every morphism $$f \colon X \to Y$$ in C we have:
$$\eta_{Y} \circ F(f) = G(f) \circ \eta_{X}$$