Let $F : \mathbb{Z} \to \mathbf{Spaces}_{\ast}$ be a function into pointed spaces equipped with a map $\sigma_n : \Sigma{F_n} \to_{\ast} F_{n+1}$ for each $n \in \mathbb{Z}$. We call $\left(F, \sigma\right)$ a prespectrum. One can check that $\left(F, \sigma\right)$ induces a $\mathbb{Z}$-indexed family of contravariant functors $\mathbf{Spaces}_{\ast} \to \mathbf{Ab}$ in the form $$ \widetilde{F}^n(X) \ := \ \mathrm{colim}_{k \in \omega}\left[ \Sigma^k{X}, F_{n +k} \right]_{\ast} $$ that satisfies the suspension and exactness axioms of a cohomology theory but maybe not the additivity axiom. Moreover, when $F$ is an $\Omega$-spectrum, in the sense that the adjuncts of $\sigma$ under the suspension-loop adjunction are homotopy equivalences, the family $\widetilde{F}$ reduces to the usual definition of cohomology arising from an $\Omega$-spectrum.
I'm curious, if $F$ is the sphere spectrum (or perhaps any suspension spectrum), then does $\widetilde{F}$ have a "natural" role in any parts of homotopy theory? If $F$ is the fibrant replacement of the sphere spectrum, then it induces stable cohomotopy. In the non-fibrant case, is $\widetilde{F}$ ever called "unstable cohomotopy"? Does it naturally show up in homotopy theory, or do we always want to fibrantly replace it first?
