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Formal unramifiedness of an arrow $f:M\rightarrow N$ in algebraic geometry or synthetic differential geometry in defined by asking the lifting problem below to have at most one solution (existence is not required). $$\array{ \mathsf{1} &\longrightarrow& M \\ \downarrow &\nearrow& \downarrow \\ D &\longrightarrow& N }$$

In SDG, I think formal unramifiedness can also defined by asking the induced arrow $TM\to P$ to the pullback below to be a monomorphism. (Recall tangent bundles are the exponentials $TM=M^D$ where $D$ is the class of zero-square "elements".)

$$\require{AMScd} \begin{CD} P @>>> TN\\ @VVV @VV{\pi^\prime}V\\ M^1 @>>{f}> N^1 \end{CD}$$

For formal smoothness, the story is dual.

Here's my problem - I can only see these definitions are equivalent in $\mathsf{Set}$, so for toposes - otherwise I don't know how to move between these descriptions.

So my question is - in what generality are these approaches to the definitions equivalent?

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