Summary: A double Roman dominating function on a graph \(G\) with vertex set \(V(G)\) is defined in [\textit{R. A. Beeler} et al., Discrete Appl. Math. 211, 23--29 (2016; Zbl 1348.05146)] as a function \(f:V(G)\rightarrow\{0,1,2,3\}\) having the property that if \(f(v)=0\), then the vertex \(v\) must have at least two neighbors assigned 2 under \(f\) or one neighbor \(w\) with \(f(w)=3\), and if \(f(v)=1\), then the vertex \(v\) must have at least one neighbor \(u\) with \(f(u)\geq 2\). The weight of a double Roman dominating function \(f\) is the sum \(\sum_{v\in V(G)}f(v)\), and the minimum weight of a double Roman dominating function on \(G\) is the double Roman domination number \(\gamma_{\mathrm{dR}}(G)\) of \(G\). A set \(\{f_1,f_2,\ldots,f_d\}\) of distinct double Roman dominating functions on \(G\) with the property that \(\sum_{i=1}^df_i(v)\leq 3\) for each \(v\in V(G)\) is called in [\textit{L. Volkmann}, ``The double Roman domatic number of a graph'', J. Comb. Math. Comb. Comput. 104, 205--215 (2018)] a double Roman dominating family (of functions)on \(G\). The maximum number of functions in a double Roman dominating family on \(G\) is the double Roman domatic number of \(G\). In this note we continue the study the double Roman domination and domatic numbers. In particular, we present a sharp lower bound on \(\gamma_{\mathrm{dR}}(G)\), and we determine the double Roman domination and domatic numbers of some classes of graphs.