# The realization functor is left adjoint to the singular functor

Proposition 2.2. The realization functor is left adjoint to the singular functor in the sense that there is an isomorphism

$$\text{hom}_\mathsf{Top}(|X|, Y) \cong \text{hom}_\mathsf{S}(X, SY)$$

which is natural in simplicial sets $$X$$ and topological spaces $$Y$$.

Proof: There are isomorphisms \begin{align} \text{hom}_\mathsf{Top}(|X|,Y) &\cong \varprojlim_{\Delta^n \to X} \text{hom}_\mathsf{Top}(|\Delta^n|, Y) \\ &\cong \varprojlim_{\Delta^n \to X} \text{hom}_\mathsf{S}(\Delta^n, S(Y) \\ &\cong \text{hom}_\mathsf{S}(X, SY). \end{align}

This is Proposition 2.2, page 7, (Simplicial Homotopy Theory, Paul G. Goerss & John F. Jardine). I understand the reasons for the first and the third isomorphisms, but I could not figure out the reason for the second isomorphism. Any help is very much appreciated.

That's the definition of $S(Y)$: its $n$-simplices are the maps $|\Delta^n|\to Y$. Note there's a Yoneda lemma in here too: $n$-simplices of a simplicial set are morphisms from $\Delta^n$.
• ... and the trick of presheaves that $F \cong \operatorname{Nat}(\mathbf{y}(-), F)$.