How to show every topological space $X$, the counit map $|\text{Sing } X | \to X$ is a weak homotopy equivalence?

Question

I wonder how to show $|\text{Sing } X |$ and $X$ are weak homotopy equivalent?

Answer 1

EDIT: Sorry for the notational difference. My $S$ is the OP’s Sing.

Suppose a space $A$ is homeomorphic to the geometric realization of some simplicial set $T$. Then maps from $A$ to $X$ are naturally in bijection with maps from $T$ to $SX$. Applying geometric realization to get a correspondingly map $A\to |SX|$, the definition of adjunction now implies that every map $A$ to $X$ factors canonically through the canonical map $|SX|\to X$.

In particular, this works when $A$ is a sphere, showing the induced map on homotopy groups is surjective, or when $A=S^n\wedge I_+$ is the space representing basepoint-fixed homotopies of maps from $S^n$, showing that the induced map is injective.