n this paper, we consider set optimization problems with respect to the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive properties of convexity and Lipschitzianity of suitable scalarizing functionals, under the same assumption on the set-valued objective mapping. We then obtain upper estimates of the limiting subdifferential of these functionals. These results, together with the scalarization properties of the functionals, allow us to obtain a Fermat rule for set optimization problems with Lipschitzian data.