The right internal Hom of pointed sets satisfies the following universal property:
\[ \mathsf{Sets}_{*}\webleft (X\rhd Y,Z\webright )\cong \mathsf{Sets}_{*}\webleft (Y,\webleft [X,Z\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\webright ) \]
That is to say, the following data are in bijection:
- Pointed maps $f\colon X\rhd Y\to Z$.
- Pointed maps $f\colon Y\to \webleft [X,Z\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$.
