I'd like to understand the proof of the exact sequence of the pair in homotopy. I'm reffering to the one I found here: https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf (p.123)
The implication I don't understand is $\text{Ker}(j_{*}) \subseteq \text{Im}(i_*)$.
I do understand the logic behind the proof but I don't get the following :
$1.$ Why $i_*[k \star u] = k \star w?$ Shouldn't always be true that if two out of four edges of the square are the costant path then the other two "edges" (representing a path) are homotopic? I don't get how $i_*$ plays a special role here.
$2.$ Related to $1$. I'd like to find an explicit homotopy of the square in order to achieve $k\star u \simeq k \star w$, as stated in the following:
Any help,hint or reference would be appreciated.