I don’t see what the problem is supposed to be; you just compose the functions witnessing the two domination relations in the obvious way:

Fix $c=u_0,\dots,u_n=d$ and $f_1,\dots,f_n\in F$ such that $\{f_i(a),f_i(b)\}=\{u_{i-1},u_i\}$  for each $i=1,\dots,n$.

Fix $e=v_0,\dots,v_m=f$ and $g_1,\dots,g_m\in F$ such that $\{g_j(c),g_j(d)\}=\{v_{j-1},v_j\}$  for each $j=1,\dots,m$.

Then the fact that $\{a,b\}$ dominates $\{e,f\}$ is witnessed by the sequence
$$e=v_0=w_{1,0},w_{1,1},\dots,w_{1,n}=v_2=w_{2,0},w_{2,1},\dots,w_{m,n}=v_m=f,$$
where
$$w_{j,i}=\begin{cases}g_j(u_i)&\text{if }g_j(c)=v_{j-1}\text{ and }g_j(d)=v_j,\\
g_j(u_{n-i})&\text{otherwise.}\end{cases}$$
We have $\{w_{j,i-1},w_{j,i}\}=\{h_{j,i}(a),h_{j,i}(b)\}$, where
$$h_{j,i}=\begin{cases}g_j\circ f_i&\text{if }g_j(c)=v_{j-1}\text{ and }g_j(d)=v_j,\\
g_j\circ f_{n+1-i}&\text{otherwise}\end{cases}$$
is in $F$ as it is closed under composition.