It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property with respect to the class $Emb$ of all embeddings of posets. This remarkable characterization would be even more useful if there were some smaller, more explicit class of embeddings $\mathcal E \subset Emb$ sufficient to check injectivity. That is,

**Questions:**

Is there a good sub-class $\mathcal E \subset Emb$ of all poset embeddings such that a poset $L$ is a complete lattice if and only if $L$ has the right lifting property with respect to all the embeddings in $\mathcal E$?

What if we restrict attention to finite posets? That is: Is there a good subclass $\mathcal E^{fin} \subset Emb^{fin}$ of all embeddings of finite posets such that a finite poset $L$ is a lattice if and only if $L$ has the right lifting property with respect to all the embeddings in $\mathcal E^{fin}$?

What is meant by "good" is a bit subjective, but to start I'd be happy with anything non-tautological. Ultimately, it would be nice to have a class $\mathcal E$ or $\mathcal E^{fin}$ which is "explicit" in some sense, so that it actually becomes easier to check that a poset $L$ is complete via lifting properties. Normally, I'd hope for $\mathcal E$ to be small, but I'm pretty sure this is not possible.

One candidate I have in mind for $\mathcal E$ would be the collection of embeddings $\{S \to S^\triangleright \mid S \text{ discrete}\} \cup \{S \to S^\triangleleft \mid S \text{ discrete}\}$ which add a new top or bottom element to a discrete poset $S$. But I suspect this is too naive, as it would mean that any $\infty$-directed and $\infty$-codirected poset is a complete lattice -- this is probably false.