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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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Use the tensor-hom adjunction. –  Qiaochu Yuan Oct 18 '12 at 21:00
    
Could you be more specific, please. I'm not familiar with tensor-hom adjunction. –  Haely Klorson Oct 18 '12 at 21:02
    
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".) –  Aaron Mazel-Gee Oct 24 '12 at 23:28

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