= Solution
Use derived <smash products of spectra> and replace a <based space> by a based <CW complex> when necessary. The reduced <represented homology theory> of $E$ is
$$
\boxed{\widetilde E_k(X)=\pi_k\bigl(E\wedge\Sigma^\infty X\bigr),\qquad k\in\mathbb Z.}
$$
Thus <represented homology theory> is covariant in $X$: a based map induces a map of <suspension spectra>, and hence a map on the <stable homotopy groups> of the displayed <smash product of spectra>. In a well-based sequential model the same definition is
$$
\widetilde E_k(X)=\operatorname*{colim}_{j}\pi_{k+j}(E_j\wedge X),
$$
with $j$ sufficiently large and transition maps given by suspension followed by the structure map.
The reduced <represented cohomology theory> is
$$
\boxed{\widetilde E^q(X)=[\Sigma^\infty X,\Sigma^qE]_{\mathrm{st}},\qquad q\in\mathbb Z.}
$$
The brackets denote maps in the <stable homotopy category>. This is contravariant in $X$, by precomposition. For the <Omega-spectrum> $E$ it is also the group of <based homotopy> classes $[X,E_q]_*$, where for negative $q$ we take $E_q=\Omega^{-q}E_0$. More generally, for $j\ge\max(q,0)$ the <Omega-spectrum> identifications give $\widetilde E^q(X)=[\Sigma^{j-q}X,E_j]_*$; the shifts and loops supply its abelian group structure. In particular
$$
\widetilde E_k(S^0)=\pi_kE,\qquad\widetilde E^q(S^0)=\pi_{-q}E.
$$
All these definitions are reduced: the basepoint itself contributes zero.
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