Definition:Composition of Functors

Definition
Let $\mathbf C, \mathbf D$ and $\mathbf E$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be (covariant) functors.

The composition of $G$ with $F$ is the functor $GF: \mathbf C \to \mathbf E$ defined by:


 * For all objects $C$ of $\mathbf C$: $\hskip{2.9cm} GF \left({C}\right) := G \left({FC}\right)$
 * For all morphisms $f: C_1 \to C_2$ of $\mathbf C$: $\quad GF \left({f}\right) := G \left({Ff}\right)$

$GF$ is said to be a composite functor.

That $GF$ in fact constitutes a functor is shown on Composite Functor is Functor.

Presentational Note
Because of the way $GF$ is defined, when a working knowledge of functors is assumed, the brackets used in defining $GF$ may be (and usually are) disposed of whenever possible.

In such situations, they can only hamper one's understanding of the expression.

Also known as
Some sources prefer to write $G \circ F$ in place of $GF$, explicitly putting the $\circ$ for composition.

Since expressions like:


 * $\left({G \circ F}\right) \left({g \circ f}\right)$

are bound to lead to confusion, the $\circ$ for composition of functors is to be suppressed on ProofWiki.