Write $sSet^{*/}$ for the category of pointed simplicial sets. There is also a full inclusion $sSet_0 \hookrightarrow sSet^{*/}$. This has a right adjoint $red : sSet^{*/} \to sSet_0$ which sends a pointed simplicial set to the subobject all whose $n$-cells have as 0-faces the given point.

Coreflection

The inclusion $sSet_0 \hookrightarrow sSet^{*/}$ into pointed simplicial sets is coreflective. The coreflector is the Eilenberg subcomplex construction in degree 1.