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How do I see that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?

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For any (reduced) cohomology theory $\tilde H^*$, not necessarily ordinary, addition in $\tilde H^r(\Sigma X)$ is induced by the pinch map $\Sigma X \to \Sigma X\ \vee \Sigma X$, using the natural isomorphism $\tilde H^r(\Sigma X \vee \Sigma X) \cong \tilde H^r(\Sigma X)\times \tilde H^r(\Sigma X)$ and functoriality. Here $\Sigma$ is the suspension functor on based spaces. From here, the conclusion follows by stability and naturality.

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