Composition of Functors is Associative

Theorem

Let $\mathbf A$, $\mathbf B$, $\mathbf C$ and $\mathbf D$ be metacategories.

Let $F: \mathbf A \to \mathbf B$, $G: \mathbf B \to \mathbf C$ and $H: \mathbf C \to \mathbf D$ be functors.

Then composition of functors is associative:

$H \paren {G F} = \paren {H G} F$

Proof

Let $A$ be an object of $\mathbf A$.

Then, solely by the definition of composite functor:

\(\ds H \paren {G F} A\)

\(=\)

\(\ds H \paren {G F A}\)

\(\ds \)

\(=\)

\(\ds H \paren {G \paren {F A} }\)

\(\ds \)

\(=\)

\(\ds H G \paren {F A}\)

\(\ds \)

\(=\)

\(\ds \paren {H G} F A\)

Then, mutatis mutandis, the same proof works for a morphism $f$ of $\mathbf A$ as well.

Hence the result.

$\blacksquare$