Definition 9.5.1.1.5 (0104): Composition of Functors

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The composition of two functors $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ is the functor $G\circ F$ where

Action on Objects. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have

\[ \webleft [G\circ F\webright ]\webleft (A\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G\webleft (F\webleft (A\webright )\webright ). \]
Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the action on morphisms

\[ \webleft (G\circ F\webright )_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{E}}\webleft (G_{F_{A}},G_{F_{B}}\webright ) \]

of $G\circ F$ at $\webleft (A,B\webright )$ is defined by

\[ \webleft [G\circ F\webright ]\webleft (f\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G\webleft (F\webleft (f\webright )\webright ). \]

Preservation of Identities

For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have

\begin{align*} G_{F_{\text{id}_{A}}} & = G_{\text{id}_{F_{A}}}\tag {functoriality of $F$}\\ & = \text{id}_{G_{F_{A}}}.\tag {functoriality of $G$} \end{align*}

Preservation of Composition

For each composable pair $\webleft (g,f\webright )$ of morphisms of $\mathcal{C}$, we have

\begin{align*} G_{F_{g\circ f}} & = G_{F_{g}\circ F_{f}}\tag {functoriality of $F$}\\ & = G_{F_{g}}\circ G_{F_{f}}.\tag {functoriality of $G$} \end{align*}

This finishes the proof.