If I have two concatenated paths $f_{1} \cdot g_{1}$ and $f_{2} \cdot g_{2}$ where $f_{1} \cong f_{2}$ and $g_{1} \cong g_{2}$, is it fair to say
$$f_{1} \cdot g_{1} \cong f_{2} \cdot g_{2}$$
by the homotopy $(f \cdot g)_{t} = (1-t)(f_{1} \cdot g_{1})+t(f_{2} \cdot g_{2})$?
Hint: consider for each $t \in I$ the (path) composition $H_t := F_t*G_t$ where $F: f_1 \simeq f_2$ and $G: g_1 \simeq g_2$. Prove that this defines a homotopy of paths: to justify the 'gluing' of the functions, use that both $F$ and $G$ fix the extrema of their corresponding paths.