For a pointed simplicial set $X$, the suspension $\Sigma X$ is calculated as the smash product $S^1\wedge X$, where $S^1=\Delta^1/\partial\Delta^1$. In the literature, there is another definition: $\Sigma X$ is calculated as the the homotopy cofiber of the canonical map $X\rightarrow *$.
Why these two definitions are equivalent?