The following question comes from Proposition 2.7, Chapter II, form the book Infinite words by D. Perrin and J.-E. Pin. I will give all the definitions after my question, so as to make this question self-contained.
Proposition 2.7. Let $S$ be a finite semigroup. Two linked pairs $(s,e)$ and $(s', e')$ of $S$ are conjugate if and only if there exists $x \in S^1$ such that $s' = sx, xe'\mathcal R e$ and $e\mathcal D e'$.
Proof. [...] Conversely, suppose there exists $x \in S^1$ such that $s' = sx, xe' \mathcal R e$ and $e \mathcal D e'$. Let $v \in S^1$ be such that $xe'v = e$. Then we have $xe' \mathcal R e = xe'v \mathcal Le'v \mathcal R e'$ and thus, by Proposition A.2.4. $h = (e'v)(xe') \mathcal H e'$. [...]
I do not understand the cited part of the proof? Maybe someone could give more details why $xe'v \mathcal L e'v$ and $e'v \mathcal R e'$ hold, and why this implies $(e'v)(xe')\mathcal He'$?
I give all the definitions. For a semigroup $S$, let $S^1 := S \cup \{1\}$ with $1s = s1 = s$ for all $s \in S$ if $S$ does not contains an identity, and $S^1 := S$ otherwise. Then we define Green's relations: \begin{align*} s \mathcal R s' & :\Leftrightarrow sS^1 = s'S^1 \\ s \mathcal L s' & :\Leftrightarrow S^1 s = S^1 s' \\ s \mathcal J s' & :\Leftrightarrow S^1 s S^1 = S^1 s' S^1 \\ s \mathcal D s' & :\Leftrightarrow s (\mathcal L \circ \mathcal R) s' \\ s \mathcal H s' & :\Leftrightarrow s\mathcal R s' \mbox{ and } s\mathcal L s' \end{align*} These are all equivalence relations, and for $s \in S$ we denote by $R(s), L(s)$ and so on the equivalence class of $s$. Some properties as taken from the book:
The relation $\mathcal R$ is stable on the left, and $\mathcal L$ is stable on the right
If $e^2 = e \in S$ and $x \in S$ with $xS^1 \subseteq eS^1$, then $ex = x$ (and similar if $S^1x \subseteq S^1 e$ we have $xe = x$).
$\mathcal L\circ \mathcal R = \mathcal R \circ \mathcal L$
(Proposition A.2.4. from the book) Let $D$ be a $\mathcal D$-class of a semigroup $S$, and let $m,m'$ and $m''$ be elements of $S$.
(1) If $m\mathcal R m'$ with $m = m'p, m' = mq$, the maps $x \mapsto xp$ and $x \mapsto xq$ define inverse bijections between $L(m)$ and $L(m')$, and these bijections preserve the $\mathcal H$-classes.
(2) One has $mm'' \in R(m) \cap L(m'')$ if and only if $R(m'')\cap L(m)$ contains an idempotent.
(3) An $\mathcal H$-class is a group if and only if it $H^2 \cap H \ne \emptyset$.
(4) If $D$ contains an idempotent, it contains at least one idempotent in each $\mathcal R$-class and in each $\mathcal L$-class.
If $S$ is finite we have
(1) For $m \in S, r,s \in S^1$ if $m = rms$ then $rm \in L(m)$ and $ms \in R(m)$.
(2) $m \mathcal D m' \Leftrightarrow m \mathcal J m'$.
(3) If $m \mathcal J m'$ and $mS^1 \subseteq m'S^1$ then $m \mathcal R m'$, and if $m \mathcal J m'$ and $S^1 m \subseteq S^1 m'$ then $m \mathcal L m'$.
Now in a semigroup two elements $s,e \in S$ are called a linked pair if $e^2 = e$ and $se = s$. Two linked pairs $s,e$ and $s',e'$ are called conjugate if there exists $x,y \in S^1$ such that $e = xy, e' = yx$ and $s' = sx$ (which also implies $s = s'y$). This conjugacy relation is an equivalence relation.
I was able to solve the issue, everything is contained in this recent question from me. In the question everything is stated in terms of monoids, without any reference to linked pairs.