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Question is in the title: How to prove that reduced suspension $\Sigma X=S^1 \wedge X$ (smash product) of pointed space $(X,x_0)$ (and $S^1$) is an $H$-cogroup?

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    At least when your spaces are all compactly generated and weak Hausdorff or something like that, the based mapping spaces satisfy the *functorial* relation $map_*(X \wedge Y,Z) \cong map_*(X,map_*(Y,Z))$. (Here, smash product is a "tensor product", and the based mapping space is the "internal hom".)2012-10-24

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