  In [{\it On the free implicative semilattice extension of a Hilbert algebra}.
Mathematical Logic Quarterly 58, 3 (2012), 188--207], Celani and Jansana give
an explicit description of the free implicative semilattice extension of a
Hilbert algebra. In this paper we give an alternative path conducing to this
construction. Furthermore, following our procedure, we show that an adjunction
can be obtained between the algebraic categories of Hilbert algebras with
supremum and that of generalized Heyting algebras. Finally, in last section we
describe a functor from the algebraic category of Hilbert algebras to that of
generalized Heyting algebras, of possible independent interest.
