Apply the construction of Remark \ref{rmk:lan-adjunction} to the embedding ${\mathbbe{\Delta}}\hookrightarrow\textup{\textsf{cat}}$ of the category of finite non-empty ordinals and order-preserving maps to define an adjunction \(\xymatrix{ \cat{Cat} \ar@<-1ex>[r]_-N \ar@{}[r]\mid-\perp & \textup{\textsf{cat}}^^\mathrm{op}}. \ar@<-1ex>[l]_-{h}}\) The right adjoint defines the nerve of a category, while the left adjoint constructs the homotopy category of a simplicial set. The counit of this adjunction is an isomorphism, so Lemma \ref{lem:counit-conditions} implies that the nerve is fully faithful. Hence, $\textup{\textsf{cat}}$ defines a reflective subcategory of the category of simplicial sets, proving the claim made in Example \ref{exs:reflective-subcats}\eqref{itm:cat-nerve-embedding}.