= Solution
Work in the <homotopy> category of <simply connected> <CW complexes>. This is one of the permitted assumptions on the <fundamental group>; it makes every coefficient system below untwisted. A <rational space> is a <simply connected> space whose <homotopy groups> are <vector spaces> over $\mathbb Q$. A <rationalization of a topological space> is a map $r:X\to X_{\mathbb Q}$ to such a space which induces
$$
\boxed{\pi_i(X_{\mathbb Q})\cong\pi_i(X)\otimes\mathbb Q\quad(i\ge2).}
$$
Its purpose includes a universal mapping property, which will be described after the construction.
An <Eilenberg–MacLane space> $K(A,n)$ has exactly one nonzero <homotopy group>, $\pi_n=A$, where $A$ is abelian when $n\ge2$. The representability theorem says $[B,K(A,n)]\cong H^n(B;A)$, naturally in $B$ and $A$. More generally, a <principal fibration> is a <homotopy pullback> of a <path-space fibration> $PC\to C$ along a map from its base to $C$; the fibre $\Omega C$ acts on the fibre paths by concatenation. For the <Postnikov tower> construction take $C=K(A,n+1)$. A <principal Eilenberg–MacLane fibration> with fibre $K(A,n)$ is the <homotopy pullback> of the <path fibration> of $K(A,n+1)$. Equivalently, its total space is the <homotopy fibre> of a map $k:B\to K(A,n+1)$. It is classified by the <cohomology class> $[k]\in H^{n+1}(B;A)$; there is no nontrivial action of the base <fundamental group> on the fibre coefficients in this principal description.
The <Postnikov tower> theorem supplies spaces $P_nX$ and compatible maps
$$
X\longrightarrow\cdots\longrightarrow P_nX\longrightarrow P_{n-1}X
\longrightarrow\cdots\longrightarrow P_1X\simeq *
$$
with $\pi_i(P_nX)=\pi_i(X)$ for $i\le n$ and zero for $i>n$. For a <simply connected> $X$, the fibre of $P_nX\to P_{n-1}X$ is $K(\pi_nX,n)$, and the fibration is principal. Its <Postnikov invariant> is
$$
k_n\in H^{n+1}(P_{n-1}X;\pi_nX),
$$
represented by $P_{n-1}X\to K(\pi_nX,n+1)$, whose <homotopy fibre> is $P_nX$. Thus the <homotopy groups> alone are not the tower data: the classes $k_n$ specify how each new group is attached. For instance a zero class gives a product with $K(\pi_nX,n)$.
State also the standard rational comparison results used in constructing the tower: $K(A,n)\to K(A\otimes\mathbb Q,n)$ is a <rational homology> equivalence; a map between <simply connected> spaces is a <rational homology> equivalence if and only if it induces isomorphisms on all <rational homotopy groups>; and a <rational homology> equivalence induces isomorphisms in <cohomology> with any constant rational vector-space coefficients. The last assertion uses the universal coefficient theorem over the <field> $\mathbb Q$. These results are used here without proof, as allowed for this essay.
Construct rational stages inductively, with $P_1^{\mathbb Q}=*$. Suppose $r_{n-1}:P_{n-1}X\to P_{n-1}^{\mathbb Q}$ has already been constructed and is a <rational homology> equivalence. Put $V_n=\pi_nX\otimes\mathbb Q$ and change coefficients in $k_n$ to obtain $\overline k_n\in H^{n+1}(P_{n-1}X;V_n)$. The comparison result gives an isomorphism
$$
r_{n-1}^*:H^{n+1}(P_{n-1}^{\mathbb Q};V_n)
\xrightarrow{\sim}H^{n+1}(P_{n-1}X;V_n).
$$
Let $k_n^{\mathbb Q}$ be its unique inverse image and define
$$
P_n^{\mathbb Q}=\operatorname{hofib}\bigl(
k_n^{\mathbb Q}:P_{n-1}^{\mathbb Q}\to K(V_n,n+1)\bigr).
$$
Equality of the pulled-back classes gives a homotopy-commuting square of classifying maps, and hence a map $r_n:P_nX\to P_n^{\mathbb Q}$ over $r_{n-1}$. On the fibre it is the map induced by $\pi_nX\to V_n$. The <homotopy> exact sequences therefore show that $r_n$ tensors all <homotopy groups> through degree $n$ with $\mathbb Q$, while the higher groups of both stages vanish. The rational comparison theorem makes $r_n$ a <rational homology> equivalence, allowing the induction to continue.
Take the <homotopy inverse limit> of the rational tower:
$$
X_{\mathbb Q}=\operatorname{holim}_n P_n^{\mathbb Q}.
$$
The convergence theorem for a <Postnikov tower> gives $X\simeq\operatorname{holim}_nP_nX$ and the compatible stage maps induce $r:X\to X_{\mathbb Q}$. To see what convergence says here, fix a <homotopy> degree $i$. From stage $n\ge i$ onwards the group and its transition maps are constant; the inverse-limit group is $\pi_iX\otimes\mathbb Q$ and the associated derived-limit obstruction is zero. Thus this construction has exactly the claimed <rational homotopy groups>. Taking a <homotopy> limit, rather than an arbitrary set-theoretic inverse limit, is part of the construction.
The rational localization theorem says that this map has the universal property
$$
\boxed{r^*:[X_{\mathbb Q},Z]\xrightarrow{\sim}[X,Z]
\quad\text{for every simply connected rational space }Z.}
$$
In other words every map from $X$ to a <rational space> factors through $r$, and the factorization is unique up to <homotopy>. This concerns <homotopy classes>, not strict uniqueness of point-set maps. The theorem can be formulated as localization of the <simply connected> <homotopy category> at <rational homotopy equivalences>; <rational spaces> are its local objects. It also implies uniqueness of <rationalization of a topological space>: apply the property to two candidate <rational spaces> to obtain inverse equivalences under $X$. In particular the construction forgets torsion in <homotopy> while retaining precisely the information needed for maps to rational targets.
The same theory extends to <nilpotent spaces>, where the <fundamental group> is nilpotent and its action on higher <homotopy groups> is nilpotent. That version requires localization of the <fundamental group> and compatible treatment of its coefficient action. The <simply connected> assumption chosen here avoids those additional issues and supplies the full construction and mapping property requested.
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