Functor

Let $C$ and $D$ be categories.
A (covariant) functor $F:C\to D$ consists of:

• A mapping on objects$$F:\mathrm{Ob}(C)\to \mathrm{Ob}(D),$$
• A mapping on morphisms$$F:{\mathrm{Hom}}_C(X,Y)\to {\mathrm{Hom}}_D(FX,FY),$$for every pair of objects $X,Y\in C$.
  These assignments satisfy:

1. Identity preservation$$F({\mathrm{id}}_X)={\mathrm{id}}_{F(X)}.$$
2. Composition preservation
   For all morphisms $f:X\to Y$ and $g:Y\to Z$,$$F(g\circ f)=F(g)\circ F(f).$$

Thus, functors preserve identities and composition.

The category of (small) categories.
Categories themselves form a category, commonly denoted $\mathbf{Cat}$:

• Objects: small categories.
• Morphisms: functors between them.
• Composition: composition of functors.
• Identities: identity functors.
  This yields a well-defined category once size issues (small vs. large categories) are handled. For large categories what we have is a 2-category.

An important relationship between functors is adjunction.

Related: natural transformation.