Currently I am reading Barratt-Puppe coexact sequence and I faced the following the problem : Let $$CX:=\frac{X\times [0,1]}{\big(\{1\}\times X\big)\cup\big(x_0\times[0,1]\big)}$$ be the reduced cone over $X$ and $x_0\in X$. Let $(Y,y_0)$ be another pointed space and $f:(X,x_0)\to(Y,y_0)$ be a pointed map. Consider the continuous map $\overline f:\overline X\to CY$ defined by $\overline f:[(x,0)]\mapsto[(f(x),0)]$ where $\overline X=\{[(x,0)]:x\in X\}$. I want to show $$CX\cup_{\overline f}CY\text{ is homeomorphic to }\Sigma X:=\frac{X ×[0,1]}{\big(X ×\{0,1\}\big)∪\big({x_0}×[0,1]\big)}.$$
Any help will be appreciated.