A functor \(\mathcal{C}\xrightarrow{F}\mathcal{D}\) between two categories.
For each object in \(\mathcal{C}\) one specifies \(F(c) \in Ob(\mathcal{D})\)
For each morphism \(c_1\xrightarrow{f}c_2\) in \(\mathcal{C}\), one specifies \(F(c_1)\xrightarrow{F(f)}F(c_2)\) in \(\mathcal{D}\)
Furthermore, two properties must be satisfied:
Identity morphisms are mapped to identity morphisms
Composition is preserved: \(F(f;g)=F(f);F(g)\)