desired shifts in the $Zb\bar b$ couplings. From Eq. (\[deltagg\]) we obtain a positive $\delta g^{b_L}$ from the gauge sector if $$\label{condbLg}
T^{3R}({\cal P}_{b_L}q^{2cp})>-1/2 \ ,$$ where $T^{3R}({\cal P}_{b_L}q^{2cp})$ is the SU(2)$_R$ charge of the composite fermion in $q^{2cp}$ that mixes with $b^{el}_L$. We also obtain a positive $\delta g^{b_R}$ from the gauge sector if $$\label{condbRg}
T^{3R}({\cal P}_{b_R}b^{cp})>0 \ .$$ Since $Y=T^{3R}+T^X$, choosing $T^{3R}$ we fix also the U(1)$_X$ charge. Eqs. (\[condbLg\]) and (\[condbRg\]), together with the condition of a gauge invariant bottom Yukawa operator, lead us to the following possible embeddings for the bottom sector $$\begin{aligned}
{\rm[}B1{\rm]}& q^{2cp}=({\bf 2},{\bf 3})_{-5/6}=
\begin{bmatrix}
 U^{cp}_2 & D^{cp'}_2 & V^{cp''}_2 \\
 D^{cp}_2 & V^{cp'}_2 & S^{cp'}_2
\end{bmatrix} \ , 
&  b^{cp}=({\bf 1},{\bf 2})_{-5/6}=\begin{bmatrix} D^{cp}_b & V^{cp'}_b\end{bmatrix} \ , \label{bembedding1}\\
{\rm[}B2{\rm]}& q^{2cp}=({\bf 2},{\bf 4})_{-4/3}=
\begin{bmatrix}
 U^{cp}_2 & D^{cp'}_2 & V^{cp''}_2 & S^{cp''}_2 \\
 D^{cp}_2 & V^{cp'}_2 & S^{cp'}_2 & T^{cp'}_2
\end{bmatrix}, 
& b^{cp}=({\bf 1},{\bf 3})_{-4/3}=\begin{bmatrix} D^{cp}_b & V^{cp'}_b & S^{cp'}_b\end{bmatrix} \ \ \ \label{bembedding2}\end{aligned}$$ where $V,S$ and $T$ are exotic fermions with $Q=-4/3,-7/3$ and $-10/3$, respectively. Is is possible to consider larger representations also. We have considered only the cases where $q^{2cp}$ and $b^{cp}$ are respectively a doublet and a singlet of SU(2)$_L^{cp}$, otherwise 