One of the most famous single-valuedness results for set-valued maps is due to Kenderov [Fund. Math., LXXXVIII (1975), pp. 61-69] and states that a monotone set-valued operator is single-valued at any point where it is lower semicontinuous. This has been extended in Christensen and Kenderov [Math. Scand., 54 (1984), pp. 70-78] to monotone maps satisfying a so-called *-property. Our aim in this work is twofold: first, to prove that the *-property assumption can be weakened, and second, to emphasize that these classical single-valuedness results for monotone operators can be obtained, in very simple way, as direct consequences of counterpart results proved for quasi-monotone operators in terms of single-directionality. © 2014 Society for Industrial and Applied Mathematics.