From this MO question I learned to tentatively think of formally unramified arrows as immersions and of formally smooth arrows as submersions.

I'm trying to semi-formally handwave myself into conviction about this but I'm getting the opposite identifications.

The setting is a commutative square as below posing a *unique* lifting problem. $$\require{AMScd} \begin{CD} P @>>> U\\ @VVV @VV{f}V\\ N @>>> X \end{CD}$$

Suppose we start in the category of topological spaces and let $P\rightarrow N$ be the inclusion of a point into an open neighborhood, and let all the other arrows also be inclusions into opens of a space $X$. Then our diagram really lives in the lattice of opens of $X$ containing the point $P$. Since we want to deal with infinitesimal neighborhoods, move to the pro-completion of this lattice and replace $N$ by some formal intersection which isn't really there to get a good notion. Now our square lives in the pro-completion with all arrows still "inclusions of opens".

The *existence* of a diagonal filler signifies that whenever an open $U$ of $X$ contains a point $P$, it also contains every infinitesimal thickening of it. (This fits perfectly with the intuition that open sets contain neighboring points.) *Uniqueness* follows from the fact all arrows are "on the nose inclusions", or, alternatively, from the fact we're living in the pro-completion of a lattice.

So open inclusions are tentatively right-orthogonal to the class of such infinitesimal thickenings of points. The same goes for open embeddings. Now since $N$ is infinitesimal, this lifting problem should only care about local proeprties of $f$. So let $f$ be a topological immersion, i.e a map which is locally-on-the-domain an embedding. I think being an immersion gives *existence* by precomposing with elements of a good open cover, but by the analogy suggested above, immersions correspond to unramified arrows whose property gives *at most* one filler.

What am I doing wrong here?

It seems my mistake must be in saying that open embeddings gives *existence*, but I don't see why they should give uniqueness (without necessarily existence) instead.