Following Evans et al. (2018a, 2018b), I use "engagement" to refer to grammatical encoding of the relative accessibility of an entity or state of affairs to the speaker and addressee. I refer to what is thereby encoded as the "engagement function". How neatly does that function map on to grammatical categories of particular languages? Here I address that question with respect to the Papuan language Ku Waru, focusing on spatial and epistemic demonstratives, and definiteness and indefinite marking. I show that forms within each of those word/morpheme classes do serve engagement functions, but in cross-cutting and partial ways. I show how the engagement function is also achieved through poetic parallelism, prosody, gaze direction and other aspects of bodily comportment. In the examples considered, the engagement function is realised through interaction between those extra-linguistic features and the grammatical ones. The main thing that is added by grammatical engagement marking is an explicit signalling of the intersubjective accord that has been achieved on other bases. I hypothesize that that is true of engagement overall, and conclude by suggesting some ways to test that hypothesis and to advance the understanding of engagement more generally.