Try this.
$U$ is faithful and reflects invertibility, as in Beck's theorem.
For every "algebra" $A$, ie object of $\textbf C$,
there are a "set" $X$ and a "function" $\eta:X\to U A$ in $\textbf D$ that is universal in the sense that
for every other "algebra" $B$ and "function" $f:X\to U B$ in $\textbf D$ there is a unique "homomorphism" $h:A\to B$ in $\textbf C$ that extends $f$ in the sense that $f=\eta;U h$.