$D,\ U,\ X$ letters will denote fermions with charges $Q=-1/3,2/3,5/3$, and the subindex will label the composite multiplet to which they belong.

The assignment for $q^{1cp}$ fixes the charges of $t^{cp}$ when we demand the top Yukawa term from Eq. (\[yukawa2\]) to be a singlet under the composite symmetries. There are two possibilities for $t^{cp}$: $$\begin{aligned}
{\rm[}T1{\rm]}& & t^{cp}=({\bf 1},{\bf 3})_{2/3}+({\bf 3},{\bf 1})_{2/3}=
\begin{bmatrix} X^{cp'}_t & U^{cp}_t & D^{cp'}_t \end{bmatrix}+
\begin{bmatrix} X^{cp''}_t \\ U^{cp'}_t \\ D^{cp''}_t \end{bmatrix}\ ,\label{tembedding1} \\
{\rm[}T2{\rm]}& & t^{cp}=({\bf 1},{\bf 1})_{2/3}=U^{cp}_t  \ .\label{tembedding2}\end{aligned}$$ We will denote with $T1$ and $T2$ the embeddings for the top sector defined by (\[tLsymmetry\],\[tembedding1\]) and (\[tLsymmetry\],\[tembedding2\]) respectively. With this assignments for $q^{1cp}$ and $t^{cp}$, the contributions from the top sector to $\delta g^{b_L}$ are very small, although there remains an irreducible contribution arising from the symmetry breaking by ${\cal L}_{mix}$ [@Casagrande:2010si]. The leading corrections to $Zb\bar b$ arise from the mixings with the bottom sector, $q^{2cp}$ and $b^{cp}$. In section \[nosym\] we will discuss how to obtain the proper corrections to $Zb\bar b$ without relying in symmetries to cancel the corrections from the top sector.

Bottom embedding {#bottomembedding}
----------------

We discuss now the embeddings of the bottom sector able to reproduce the 