How is this lemma used?

Question

The lemma is given below:

And the use of is in the following line:

Where $f'$ is the inverse of $f$, and $f: X \rightarrow Y$ and $g: Y \rightarrow Z$.

I do not understand how this lemma is used in this line, could anyone explain this for me please?

Answer 1

The function $f'$ is not an inverse of $f$ as a function, but as a homotopy equivalence; therefore, it is not true that $f \circ f' = 1_Y$ but rather $f \circ f' \simeq 1_Y$. The lemma then allows you to say that $$ g\circ(f \circ f') \simeq g\circ (1_Y) = g, $$ and then that $$ (g\circ(f \circ f'))\circ g' \simeq (g) \circ g' \simeq 1_Z.$$ The same holds for the second line.