 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 $E$ isThus 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 iswith $j$ sufficiently large and transition maps given by suspension followed by the structure map.

The reduced represented cohomology theory isThe 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 $[X,E_q{]}_{\ast }$, 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 ${\tilde{E}}^q(X)=[{\Sigma }^{j-q}X,E_j{]}_{\ast }$; the shifts and loops supply its abelian group structure. In particularAll these definitions are reduced: the basepoint itself contributes zero.