Here is some intuition on why $-\rhd Y$ fails to be a left adjoint. Item 4 of Proposition 5.3.1.1.7 states that we have a natural bijection
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\rhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}}\webleft (|X|,\mathsf{Sets}_{*}\webleft (Y,Z\webright )\webright ), \]
so it would be reasonable to wonder whether a natural bijection of the form
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\rhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,\textbf{Sets}_{*}\webleft (Y,Z\webright )\webright ), \]
also holds, which would give $-\rhd Y\dashv \textbf{Sets}_{*}\webleft (Y,-\webright )$. However, such a bijection would require every map
\[ f\colon X\rhd Y\to Z \]
to satisfy
\[ f\webleft (x_{0}\rhd y\webright )=z_{0} \]
for each $x\in X$, whereas we are imposing such a basepoint preservation condition only for elements of the form $x\rhd y_{0}$. Thus $\textbf{Sets}_{*}\webleft (Y,-\webright )$ can’t be a right adjoint for $-\rhd Y$, and as shown byItem 3 of Proposition 5.4.1.1.7, no functor can.1
1The functor $\textbf{Sets}_{*}\webleft (Y,-\webright )$ is instead right adjoint to $-\wedge Y$, the smash product of pointed sets of Definition 5.5.1.1.1. See Item 2 of Proposition 5.5.1.1.10.
