  Assume that $\mathbb F$ is an algebraically closed field and $q$ is a nonzero
scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson
algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra generated by
$A,B, C$ and the relations state that each of $$ A+\frac{q BC-q^{-1}
CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q
AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $\triangle_q$. The universal DAHA
$\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb
F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations state that
\begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\
\hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\
t_0t_1t_2t_3=q^{-1}. \end{gather*} It was given an $\mathbb F$-algebra
homomorphism $\triangle_q\to \mathfrak H_q$ that sends \begin{eqnarray*} A
&\mapsto & t_1 t_0+(t_1 t_0)^{-1}, \\ B &\mapsto & t_3 t_0+(t_3 t_0)^{-1}, \\ C
&\mapsto & t_2 t_0+(t_2 t_0)^{-1}. \end{eqnarray*} Therefore any $\mathfrak
H_q$-module can be considered as a $\triangle_q$-module. Let $V$ denote a
finite-dimensional irreducible $\mathfrak H_q$-module. In this paper we show
that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard
triples on all composition factors of the $\triangle_q$-module $V$.
