The orthogonal combination $q_L^{el}=(q_L^{1el} \ \sin\gamma+q_L^{2el} \ \cos\gamma)$ remains massless up to mixings with the composite sector, this $q_L^{el}$ gives rise to the SM quark doublet after elementary/composite diagonalization. We assume the elementary mass scale to be much larger than the composite scale, of the order of the UV cut-off of the elementary sector.

A symmetry for top contributions to $Zb\bar b$ {#sym}
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To protect the $Zb_L\bar b_L$ coupling from too large corrections arising from the mixings with $q^{1cp}$, we can invoke a $P_{LR}$ symmetry [@Agashe:2006at] (it exchanges $L\leftrightarrow R$), assigning the following charges for ${\cal P}_{b_L}q^{1cp}$, the component of $q^{1cp}$ that mixes with $b^{el}_L$ in Eq. (\[Lmix\]): $$\label{ZbLbL}
T^L({\cal P}_{b_L}q^{1cp})=T^R({\cal P}_{b_L}q^{1cp})=1/2 \ , \qquad T^{3L}({\cal P}_{b_L}q^{1cp})=T^{3R}({\cal P}_{b_L}q^{1cp})=-1/2 \ .$$ Thus we obtain: $$\begin{aligned}
\label{tLsymmetry}
q^{1cp}=({\bf 2},{\bf 2})_{2/3}=
\begin{bmatrix}
 X^{cp'}_1 & U^{cp}_1 \\
 U^{cp'}_1 & D^{cp}_1 
\end{bmatrix} \ ,\end{aligned}$$ where the fields $U_1$ and $D_1$ have electric charges $Q=2/3$ and $Q=-1/3$ respectively, and mix with $q^{el}_L$ in ${\cal L}_{mix}$, the prime fields $U'_1$ and $X'_1$ are the custodians, with charges $Q=2/3$ and $Q=5/3$. In the rest of the paper, to fix the notation for the components of the composite multiplets, we will use ’ superindices for the custodial fermions, 