Functor

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Definition

Suppose $\mathcal{C},\mathcal{D}$ are categories. A functor (also called covariant functor) $\mathcal{F}$ from $\mathcal{C}$ to $\mathcal{D}$ comprises the following data:

• Object level mapping: A mapping $\mathcal{F}:\mathrm{O}\mathrm{b}\mathcal{C}\to \mathrm{O}\mathrm{b}\mathcal{D}$.
• Morphism level mapping: For any $A,B\in \mathcal{C}$, a mapping $\mathcal{F}:\mathcal{C}(A,B)\to \mathcal{D}(FA,FB)$.

satisfying the following condition:

• It preserves the identity morphism: For any $A\in \mathcal{C}$, $\mathcal{F}({\mathrm{i}}_{\mathrm{d}})={\mathrm{i}}_{\mathrm{d}}$.
• It preserves composition: For any $A,B,C\in \mathcal{C}$, and $f\in \mathcal{C}(A,B),g\in \mathcal{C}(B,C)$, we have $F(g\circ f)=Fg\circ Ff$.

There is a related notion of contravariant functor. Note that a contravariant functor between two categories is not a functor between the categories. To emphasize that a functor is a functor and not a contravariant functor, we use the term covariant functor.