Represented cohomology theory (${\tilde{E}}^{\ast }(X)$)

A topological spectrum $E$ represents the reduced generalized cohomology theory ${\tilde{E}}^q(X)=[{\Sigma }^{\infty }X,{\Sigma }^{q}E{]}_{\mathrm{st}}$. For an Omega-spectrum this equals $[X,E_q{]}_{\ast }$, taking $E_q={\Omega }^{-q}{E}_0$ for negative $q$. It is contravariant in $X$.