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We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if $E$ and $F$ are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint Dunford-Pettis operator $T^{\text{'}}$ from $F^{\text{'}}$ into $E^{\text{'}}$ if, and only if, the norm of $E^{\text{'}}$ is order continuous or $F^{\text{'}}$ has the Schur property. As a consequence we show that, if $E$ and $F$ are two Banach...