Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.
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Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto X\rhd Y$ define functors
\[ \begin{array}{ccc} X\rhd -\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\rhd Y\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\rhd -_{2}\colon \mkern -15mu & \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]
In particular, given pointed maps
\begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (A,a_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (B,b_{0}\webright ), \end{align*}the induced map
\[ f\rhd g\colon X\rhd Y\to A\rhd B \]is given by
\[ \webleft [f\rhd g\webright ]\webleft (x\rhd y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright )\rhd g\webleft (y\webright ) \]for each $x\rhd y\in X\rhd Y$.
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Adjointness I. We have an adjunctionwitnessed by a bijection of sets
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\rhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (Y,\webleft [X,Z\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\webright ) \]natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, where $\webleft [X,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$ is the pointed set of Definition 5.4.2.1.1.
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Adjointness II. The functor
\[ -\rhd Y\colon \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]
does not admit a right adjoint.
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Adjointness III. We have a ${\text{忘}}$-relative adjunction witnessed by a bijection of sets
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\rhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}}\webleft (|X|,\mathsf{Sets}_{*}\webleft (Y,Z\webright )\webright ) \]natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
