Example 8.4.1.1.4 (0103): Identity Functors

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The identity functor of a category $\mathcal{C}$ is the functor $\text{id}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ where

Action on Objects. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have
\[ \text{id}_{\mathcal{C}}\webleft (A\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A. \]
Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the action on morphisms
\[ \webleft (\text{id}_{\mathcal{C}}\webright )_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \underbrace{\textup{Hom}_{\mathcal{C}}\webleft (\text{id}_{\mathcal{C}}\webleft (A\webright ),\text{id}_{\mathcal{C}}\webleft (B\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )} \]

of $\text{id}_{\mathcal{C}}$ at $\webleft (A,B\webright )$ is defined by

\[ \webleft (\text{id}_{\mathcal{C}}\webright )_{A,B} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )}. \]

Preservation of Identities

We have $\text{id}_{\mathcal{C}}\webleft (\text{id}_{A}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{A}$ for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$ by definition.

Preservation of Compositions

For each composable pair $A\xrightarrow {f}B\xrightarrow {g}B$ of morphisms of $\mathcal{C}$, we have

\begin{align*} \text{id}_{\mathcal{C}}\webleft (g\circ f\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\mathcal{C}}\webleft (g\webright )\circ \text{id}_{\mathcal{C}}\webleft (f\webright ). \end{align*}

This finishes the proof.