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For $D_{n+1}=[D_n\to D_n]$ with recursively lifted embedding-projection pairs, form coherent sequences $x_n=p_n{x}_{n+1}$, ordered componentwise. Stage projections $P_n$ and embeddings $E_n$ satisfy $P_n{E}_n=\mathrm{id}$, $E_n{P}_n\uparrow \mathrm{id}$. A coherent family $f_n\in [D_n\to D_n]$ induces $f={\sup }_n{E}_n{f}_n{P}_n$; conversely $f_n=P_{n}f{E}_n$. These constructions are inverse. Removing the initial coordinate of the coherent sequence identifies this function-space limit with the original domain limit.

An embedding-projection pair consists of continuous maps $e:D\to E$, $p:E\to D$ with $pe={\mathrm{id}}_D$ and $ep\le {\mathrm{id}}_E$ pointwise. Thus $e$ is an order embedding and $ep$ approximates an element of the larger domain from below. Such maps are automatically strict: applying the inequality at ${\perp }_E$ gives $e({\perp }_D)\le ep({\perp }_E)\le {\perp }_E$, and applying $p$ then gives $p({\perp }_E)={\perp }_D$.

Suppose $({\varphi }_{n-1},{\psi }_{n-1})$ is such a pair. Composition with fixed continuous maps is continuous on the pointwise function space of complete partial orders, so both proposed lifted maps are continuous. Their composite on the smaller function space isWriting $q={\varphi }_{n-1}{\psi }_{n-1}\le \mathrm{id}$, the other composite is ${\varphi }_n{\psi }_n(g)=qgq\le g$: for every $x$, $q(g(qx))\le g(qx)\le g(x)$ by monotonicity. Thus the recursion produces embedding-projection pairs at every stage.

Here is an explicit inverse-limit solution of the reflexive domain equation. Formordered coordinatewise. The bottom sequence is coherent by strictness. A directed family has coordinatewise suprema, and continuity of each ${\psi }_n$ preserves the coherence condition. Thus $D_{\infty }$ is a pointed complete partial order.

Write ${\varphi }_{m,n}:D_m\to D_n$ for the composite embeddings when $m\le n$, and ${\psi }_{m,n}:D_n\to D_m$ for the corresponding composite projections, taking both as identities if $m=n$. Define $P_n:D_{\infty }\to D_n$ by $P_n(x)=x_n$, and define $E_n:D_n\to D_{\infty }$ byThe pair identities make this sequence coherent, and its coordinates are continuous. They also give $P_n{E}_n={\mathrm{id}}_{D_n}$ and $E_n{P}_n\le {\mathrm{id}}_{D_{\infty }}$. Put $Q_n=E_n{P}_n$. Since $E_n=E_{n+1}{\varphi }_n$ and $P_n={\psi }_n{P}_{n+1}$,At coordinate $m$, $Q_{n}x$ equals $x_m$ once $n\ge m$, soThis exhibits the required embedding-projection pair for every finite stage.

To prove the function-space isomorphism, observe that $x_{n+1}\in D_{n+1}$ is itself a continuous function $f_n:D_n\to D_n$. Coherence saysConversely such a coherent function family determines an element of $D_{\infty }$, with initial coordinate $x_0={\psi }_0(f_0)$. Define its action on the limit byThe maps inside this supremum increase: substituting coherence expresses the $n$th one as $E_{n+1}{q}_n{f}_{n+1}{q}_n{P}_{n+1}$ with $q_n={\varphi }_n{\psi }_n\le \mathrm{id}$, which is below the next map. The supremum is continuous by the function-space result in Question 3.

In the other direction, for a continuous $F:D_{\infty }\to D_{\infty }$, define $f_n=P_{n}F{E}_n$. These satisfy the same coherence equation. Hence define $\mathcal{B}(F)\in D_{\infty }$ byFor a coherent family, restricting ${\sup }_n{E}_n{f}_n{P}_n$ back to stage $m$ gives $f_m$: every term with $n\ge m$ restricts exactly to ${\psi }_{m,n}{f}_n{\varphi }_{m,n}=f_m$, and the earlier terms are below it. Thus $\mathcal{B}\mathcal{A}$ is the identity.

Conversely,For the last equality, $Q_n\uparrow \mathrm{id}$ and continuity give, at every $x$, ${\sup }_n{Q}_{n}F(Q_{n}x)={\sup }_{n,m}{Q}_{n}F(Q_{m}x)=F(x)$; the diagonal is cofinal because the double family increases in both indices. Both $\mathcal{A}$ and $\mathcal{B}$ are continuous, by coordinatewise evaluation, continuous composition and directed suprema. Thereforeas complete partial orders with continuous maps, not merely as sets. This supplies the extensional reflexive object needed for a beta-eta model, together with all the stated finite-stage pairs.