Describing the homotopy explicitly.

Question

Here is the question I am trying to solve:

Let $X$ be a based space, and let $PX = \{ \beta: I \to X | \beta(0) = *\}.$ Show that $p_1: PX \to X$ by $p_1(\beta) = \beta(1)$ is a based fibration.

I am confused about what should be the exact homotopy that I should write, here is a trial:

We have the following commutative diagram:

and here is a trial to the homotopy I should write:

$H'(y,s)(t) = \begin{cases} f(y) \left( \frac{2t}{2-s} \right) & 0 \leq t \leq 1- \frac{s}{2},\\ H(y, 2(t-1)+s) & 1- \frac{s}{2} \leq t \leq 1. \end{cases}$

Is this a correct homotopy or not? Can it be written and thought of in a more organized and systematic way.

Edit for Anne:

You draw a square like below and then you draw your homotopy as a line joining s(down) with s(up) then you determine the coordinates of the points on both the s axes and then you calculate the equation of the line joining the points on the x-axes which help in determining the point at which your homotopy changes its definition.

EDIT 2 for Anne :

Is this what you meant Anne?

Answer 1

I checked your homotopy and I think it is correct. The following seems to be correct also (?), and simpler: $$H'(y,s)(t) = \begin{cases} f(y)(s+t) & s+t\le1,\\ H(y,s+t-1) & s+t\ge1. \end{cases}$$