On p. 12 of Introduction to Lattices and Order by Davey and Priestley, the authors give a 1-paragraph description of Category Theory, and then write:
We do not have sufficient need to call on the theory of categories to warrant setting up its formalism here, but it would be wrong not to acknowledge its subliminal influence.
I find the authors' choice here hugely disappointing. Can someone recommend an introduction to order theory that does use category theory in its presentation?
It's rather idiosyncratic in the topics it covers, but you might get something out of Fong and Spivak's Seven Sketches in Compositionality. It's a textbook on applied category theory, but it begins with preorders and only later generalises to categories. The book has a tendency to try and build suspense, so the connection to categories isn't made explicit at first (it doesn't even give the defintion of a category until several chapters in), but the connection is very much there and gets revealed more explicitly later on.