Want to form the mapping cone of a map $f: X\rightarrow Y$ in the homotopy category.
I am hoping that some one can give easy examples to show that mapping cone $Y \cup_f CX$ does not dependent on f functorially and comments on when one should be careful about this.
I read this from Nilpotence and stable homotopy theory 2. In page 6, the notation and conventions section.
Thanks!