A notion of Hochschild cohomology HH*(๐) of an abelian category ๐ was defined by Lowen and Van den Bergh (Adv. Math. 198 (2005), 172โ221). They also showed theย existence of a characteristic morphism ฯ from the Hochschild cohomology of ๐ into the graded centre โจ*(Db(๐)) of the bounded derived category ofย ๐. An element cโHH2(๐) corresponds to a first-order deformation ๐c of ๐ (Lowen and Van den Bergh, Trans. Amer. Math. Soc. 358 (2006), 5441โ5483). The problem of deforming an object MโDb(๐) to Db(๐c) was treated by Lowen (Comm. Algebra 33 (2005), 3195โ3223). In this paper we show that the element ฯ(c)MโExt๐2(M,M) is precisely the obstruction to deforming M to Db(๐c). Hence, this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a โderived deformation theoryโ.