In Cocyclic maps and coevaluation subgroups by Lim, a map $f:X \to A$ is said to be cocyclic if there exists a map $\phi$ such that the diagram is homotopy commutative.

Is the projection $ f: S^2 \times S^3 \to S^2 $ cocyclic?

Where $S^n$ is the $n$-sphere.

I'm aware of the well known facts about the co-$H$-space structure on $S^n$ for some $n$, but I'm not sure about the product spheres. I think- if I'm not wrong- that using cup-products we can show the answer is ** No**, but I'm not sure how to put it together.