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Assuming that the diagonal map $X\rightarrow X\times X$ is a cofibration. Is it true that the diagonal map $\Sigma X\rightarrow \Sigma X\times \Sigma X$ is a cofibration? (Where $\Sigma X$ is the reduced suspension of $X$.)

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Yes. See Gaunce Lewis' paper

http://www.ams.org/journals/tran/1982-273-01/S0002-9947-1982-0664034-8/

By definition $X$ is locally equiconnected (LEC) iff the diagonal map $X \to X \times X$ is a cofibration (of unbased spaces). It follows that $X \times I$ is LEC, which implies that $\Sigma X$ is LEC, by the Dyer-Eilenberg adjunction theorem for LEC spaces. See Theorem 2.3 in Lewis' paper.

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