= Solution
With the Euclidean inner product used above, the simple <coroots> are
$$
\boxed{\alpha_1^\vee=\varepsilon_1-\varepsilon_2,\qquad
\alpha_2^\vee=\varepsilon_2-\varepsilon_3,\qquad
\alpha_3^\vee=\varepsilon_3.}
$$
Indeed, if a positive root is $\beta=\sum_i n_i\alpha_i$, then
$$
\beta^\vee=\sum_i n_i\frac{(\alpha_i,\alpha_i)}{(\beta,\beta)}\alpha_i^\vee,
$$
whose coefficients are nonnegative. The negative roots give the negatives of these combinations, so the displayed coroots form a <fundamental system of a root system> for the <dual root system>.
Duality reverses root lengths. The dual of $C_3$ is therefore the <B3 root system>: its <Dynkin diagram> is again a three-node chain with a double final edge, but its arrow points toward the now-short root $\alpha_3^\vee$.
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