  The present work deals with the canonical map of smooth, compact complex
surfaces of general type in a polarization of type $(1,2,2)$ on an abelian
threefold. A natural and classical question is whether the canonical system of
such surfaces is very ample in the general case. In this work, we provide a
positive answer to this question. First, we describe the structure of the
canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian
threefold which are bidouble cover of principal polarizations. Then, we study
the general behavior of the canonical map of general ample surfaces
$\mathcal{S}$, yielding a $(1,2,2)$-polarization on an abelian threefold $A$
which is isogenous to a product. By combining these descriptions, we show that
the canonical map yields a holomorphic embedding when $A$ and $\mathcal{S}$ are
both sufficiently general. It follows, in particular, a proof of the existence
of canonical irregular surfaces in $\mathbb{P}^5$ with numerical invariants
$p_g = 6$, $q = 3$ and $K^2 = 24$.
