I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper mean from a categorial perspective, I arrived at the following simple construction.
Let $P$ be a poset with $0$, not necessarily directed. Write $\nabla$ for the relation $P\times P\to P$, where $(a_1,a_2)\nabla b$ if and only if $a_1,a_2\leq b$. Then $(P,\nabla)$ is a semigroup in the monoidal category $(\mathbf{Rel},\times,1)$. However, it is not a monoid: the arrow $0:1\to P$ does not satisfy the triangle axiom strictly, for $a\in P$ we only have $a\in\nabla(0,a)$.
Question: What is the approriate categorial setting to describe such "monoids with lax unit"?