Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function.

I would like to know whether there is a name and perhaps a different characterizations of such functions $\phi$ that satisfy the following condition.

For each $p \in P$ the restriction $\phi\restriction_p : p\downarrow \to \phi(p)\downarrow$ has a right adjoint section $\phi^p$. By this I mean that $\phi^p$ is an order-preserving function $\phi^p : \phi(p)\downarrow \to p\downarrow$ that satisfies $\phi^p \circ \phi\restriction_p \geq \text{id}$ and $\phi\restriction_p \circ \phi^p = \text{id}$.

By $p\downarrow$ I mean the set $\left\{ p' \in P\ |\ p' \leq p\right\}$.

An alternative (only barely) characterizations of these maps is that for each $p \in P$ and $q \in Q$ such that $q \leq \phi(p)$ the set of elements $p' \in P$ below $p$, i.e. $p' \leq p$ that map into $q$ has a top element $u(p,q)$ and if $q_1 \leq q_2 \leq \phi(p)$ then $u(p,q_1) \leq u(p,q_2)$.


These are just (Grothendieck) fibrations as specialised to partially ordered sets and order preserving functions.


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