The models of some computational calculi are in a correspondence with Cartesian Closed Categories with an object $U$ that has some relationship to its exponential object $U^U$ e.g. a retraction between $U$ and $U^U$ gives $\lambda$-algebra whereas those with enough points and $U \cong U^U$ admitting extensionality
One possible relationship between $U$ and $U^U$ is (weak) point-surjectivity which is defined as there existing a morphism $\phi : U \rightarrow U^U$ such that for all points $p: 1 \rightarrow U^U$, there exists a point $u : 1 \rightarrow U$ and $\phi \circ u = p$.
This relationship is relevant to $\lambda$-calculi as the $F: D \rightarrow D^D$ sided arrow in the retraction for lambda models and in the isomorphism can be seen to be point-surjective. Furthermore point-surjectivity appears in Lawvere's Fixed Point Theorem which can be used to derive the First-Fixed-Point theorem for the (I have written a derivation here however it has been rushed so apologies for mistakes).
A question that I have been contemplating is whether point-surjectivity corresponds to any existing class of automata or combinatorial calculi. Longo and Moggi in "A Category-Theoretic Characterization of Functional Completeness" outline the categorical models associated with combinatory algebras however it is not immediately obvious to me whether we point-surjectivity corresponds to this formulation however my intuition says otherwise. I have managed to derive several combinators in an applicative structure induced by a point-surjective $F: U \rightarrow U^U$ in the standard way as outlined in Chapter 5 of Barendregt's "The Lambda Calculus: It's Syntax and Semantics". These are the Mockingbird combinator $\textbf{M} x = x x$, the Identity combinator $\textbf{I} x = x$, and the combinator $\textbf{F} x y = y$. Derivations can be found here. I am unsure if this is an exhaustive list of the combinators that can be derived or if this set has any meaningful interpretation.
Any guidance or insights would be greatly welcomed!
