Inverse-limit solution of the reflexive domain equation ($D_{\infty }\cong [D_{\infty }\to D_{\infty }]$)

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.