Let
$A$
be a finite alphabet and
$f:~A^{\ast }\rightarrow A^{\ast }$
be a morphism with an iterative fixed point
$f^{{\it\omega}}({\it\alpha})$
, where
${\it\alpha}\in A$
. Consider the subshift
$({\mathcal{X}},T)$
, where
${\mathcal{X}}$
is the shift orbit closure of
$f^{{\it\omega}}({\it\alpha})$
and
$T:~{\mathcal{X}}\rightarrow {\mathcal{X}}$
is the shift map. Let
$S$
be a finite alphabet that is in bijective correspondence via a mapping
$c$
with the set of non-empty suffixes of the images
$f(a)$
for
$a\in A$
. Let
${\mathcal{S}}\subset S^{\mathbb{N}}$
be the set of infinite words
$\mathbf{s}=(s_{n})_{n\geq 0}$
such that
${\it\pi}(\mathbf{s}):=c(s_{0})f(c(s_{1}))f^{2}(c(s_{2}))\cdots \in {\mathcal{X}}$
. We show that if
$f$
is primitive,
$f^{{\it\omega}}({\it\alpha})$
is aperiodic, and
$f(A)$
is a suffix code, then there exists a mapping
$H:~{\mathcal{S}}\rightarrow {\mathcal{S}}$
such that
$({\mathcal{S}},H)$
is a topological dynamical system and
${\it\pi}:~({\mathcal{S}},H)\rightarrow ({\mathcal{X}},T)$
is a conjugacy; we call
$({\mathcal{S}},H)$
the suffix conjugate of
$({\mathcal{X}},T)$
. In the special case where
$f$
is the Fibonacci or Thue–Morse morphism, we show that the subshift
$({\mathcal{S}},T)$
is sofic, that is, the language of
${\mathcal{S}}$
is regular.