@JonathanBeardsley As a functor from pointed simplicial sets to simplicial sets, $\Omega$ has a left adjoint which sends a simplicial set $A$ to the pushout of the span $\Delta^0 \star A \leftarrow A \rightarrow \Delta^0$ (which you might write as $(\Delta^0 \star A)/A$). This explicit description follows from the definition of $\Omega(X,x) = Hom_X^L(x,x)$ as the fibre of $x \backslash X \rightarrow X$ over $x$ and the usual join--slice adjunction.