  Let $H$ be a torsion-free $\delta$-hyperbolic group with respect to a finite
generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of
$H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there
is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to
$W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a
computable constant depending only on $\delta$, $\sharp{S}$ and $\sum_{i=1}^n
|a_i|$.
  As a corollary, we deduce that there exists a computable constant
$\mathcal{C}=\mathcal{C}(\delta, \sharp S)$ such that, for any endomorphism
$\phi$ of $H$, if $\phi(h)$ is conjugate to $h$ for every element $h\in H$ of
length up to $\mathcal {C}$, then $\phi$ is an inner automorphism.
  Another corollary is the following: if $H$ is a torsion-free conjugacy
separable hyperbolic group, then $\text{\rm Out}(H)$ is residually finite.
  When particularizing the main result to the case of free groups, we obtain a
solution for a mixed version of the classical Whitehead's algorithm.
