It is a standard fact that smashing with a fixed spectrum $Z$ preserves cofiber sequences. So if I have a cofiber sequence $$X \xrightarrow{f} Y \rightarrow C\_f$$ then there is also a cofiber sequence $$Z \wedge X \rightarrow Z \wedge Y \rightarrow Z \wedge C\_f$$


If more generally I have a map $Z \xrightarrow{g} W$, is there any formula for the cofiber of the map $$Z \wedge X \xrightarrow{g \wedge f} W \wedge Y$$ in terms of $C\_f$ and $C\_g$? (The above discussion corresponding to $g = \mathrm{id}\_Z$).



The homotopy cofiber of the smash product of two maps of spectra
By factoring $g\wedge f$ as $X\wedge f$ followed by $g\wedge Y$ you see that it's in the middle of a cofiber sequence $Z\wedge C\_f\to \ ?\to C\_g\wedge Y$. Similarly it's in the middle of a cofiber sequence $C\_g\wedge X\to\ ?\to W\wedge C\_f$. It's also in the middle of a cofiber sequence $(Z\wedge C\_f)\vee (C\_g\wedge X)\to \ ?\to C\_g\wedge C\_f$. We could probably come up with more. But no formula in the sense that I think you mean. It may be instructive to think of the special case where $X$ and $W$ are both contractible.


