Abstract: The category consists of po-monoids and morphisms ϕ: A→B which satisfy ϕ(a)ϕ(a′) ≤ ϕ(aa′) and preserve the order. Among the monics are maps called embeddings and the injectives are with respect to the embeddings.
Note: This started out as a note by Lambek from 2000, which turned out to have a gap. We tried to fill the gap and had to change some of the definitions somewhat, but succeeded in finding an injective envelope.