Functor

Let $C$ and $D$ be categorys. A functor $F$ from $C$ to $D$ is a mapping that

• associates each object $X\in ob(C)$ an object $F(X)\in ob(D)$.

• associates each morphism f : X → Y in $C$ to a morphism F ( f ) : F ( X ) → F ( Y ) in $D$ such that the following two conditions hold:

• -- $F(id_X)=id_{F(X)}$ for every $X\in ob(C)$.

• -- $F(g\circ f)=F(g)\circ F(f)$ for all morphisms $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ in $C$.

That is, functors must preserve identity morphisms and composition of morphisms.

As a result, this defines a category of categories and functors – the objects are categories, and the morphisms (between categories) are functors.

An important relationship between functors is adjunction.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es