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.

This is justified by the definition above, and the result Composition of Functors is Associative.

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.