Here is some intuition on why $X\lhd -$ fails to be a left adjoint. Item 4 of Proposition 5.3.1.1.7 states that we have a natural bijection
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}}\webleft (|Y|,\mathsf{Sets}_{*}\webleft (X,Z\webright )\webright ), \]
so it would be reasonable to wonder whether a natural bijection of the form
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (Y,\textbf{Sets}_{*}\webleft (X,Z\webright )\webright ), \]
also holds, which would give $X\lhd -\dashv \textbf{Sets}_{*}\webleft (X,-\webright )$. However, such a bijection would require every map
\[ f\colon X\lhd Y\to Z \]
to satisfy
\[ f\webleft (x\lhd y_{0}\webright )=z_{0} \]
for each $x\in X$, whereas we are imposing such a basepoint preservation condition only for elements of the form $x_{0}\lhd y$. Thus $\textbf{Sets}_{*}\webleft (X,-\webright )$ can’t be a right adjoint for $X\lhd -$, and as shown byItem 3 of Proposition 5.3.1.1.7, no functor can.1
1The functor $\textbf{Sets}_{*}\webleft (X,-\webright )$ is instead right adjoint to $X\wedge -$, the smash product of pointed sets of Definition 5.5.1.1.1. See Item 2 of Proposition 5.5.1.1.9.
