$Dist Lat$ is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so has all the usual properties of such categories. The free distributive lattice on a set$X$ is the set of finite antichains of the finite power set$\mathcal{P}_{fin}$ of $X$; a collection $\mathcal{C}$ of subsets of $X$ is an antichain if, whenever $A \subseteq B$ for $A, B \in \mathcal{C}$, we have $A = B$ (that is, no two distinct elements of $\mathcal{C}$ may be comparable). Note that $\mathcal{P}_{fin}X$ is the free semilattice on $X$; we may interpret an element of it as a finitary join of elements of $X$. Then we interpret an element of $\mathcal{P}_{fin}\mathcal{P}_{fin}X$ as a finitary meet of elements of $\mathcal{P}_{fin}X$. However, if two of the elements of $\mathcal{P}_{fin}X$ that appear in this meet are comparable, then we need only the smaller of them, so we take only the irredundant elements of $\mathcal{P}_{fin}\mathcal{P}_{fin}X$. (We could equally well take joins of meets instead of meets of joins; then we keep only the larger of two meets that appear in a given join.)