- Objects. We have
\[ \text{Obj}\webleft (X_{\mathsf{indisc}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \]
- Morphisms. For each $A,B\in \text{Obj}\webleft (X_{\mathsf{indisc}}\webright )$, we have
\begin{align*} \textup{Hom}_{X_{\mathsf{disc}}}\webleft (A,B\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft [A\webright ]\to \webleft [B\webright ]\webright\} \\ & \cong \text{pt}. \end{align*}
- Identities. For each $A\in \text{Obj}\webleft (X_{\mathsf{indisc}}\webright )$, the unit map
\[ \mathbb {1}^{X_{\mathsf{indisc}}}_{A} \colon \text{pt}\to \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (A,A\webright ) \]
of $X_{\mathsf{indisc}}$ at $A$ is defined by
\[ \text{id}^{X_{\mathsf{indisc}}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft [A\webright ]\to \webleft [A\webright ]\webright\} . \]
- Composition. For each $A,B,C\in \text{Obj}\webleft (X_{\mathsf{indisc}}\webright )$, the composition map
\[ \circ ^{X_{\mathsf{indisc}}}_{A,B,C} \colon \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (B,C\webright ) \times \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (A,B\webright ) \to \textup{Hom}_{X_{\mathsf{indisc}}}\webleft (A,C\webright ) \]
of $X_{\mathsf{disc}}$ at $\webleft (A,B,C\webright )$ is defined by
\[ \webleft (\webleft [B\webright ]\to \webleft [C\webright ]\webright )\circ \webleft (\webleft [A\webright ]\to \webleft [B\webright ]\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\webleft [A\webright ]\to \webleft [C\webright ]\webright ). \]