This is basically a restatement of [this question](https://mathoverflow.net/questions/248002/when-are-descriptions-of-formal-unramifiedness-smoothness-via-lifting-properties).


Two arrows $f,g$ are orthogonal, i.e satisfy $f\perp g$, iff the square below is a pullback


$$\require{AMScd} \begin{CD}
 \mathsf C(B,X) @>{f^\ast}>> \mathsf C(A,X)\\ @V{g\_\ast}VV @VV{g\_\ast}V\\
 \mathsf C(B,Y) @>>{f^\ast}> \mathsf C(A,Y).
 \end{CD}$$


Suppose now $\mathsf C$ is closed and we replace the hom-sets by internal homs.
$$\require{AMScd} \begin{CD}
 [B,X] @>{[f,-]}>> [A,X]\\ @V{[-,g]}VV @VV{[-,g]}V\\
 [B,Y] @>>{[f,-]}> [A,Y].
 \end{CD}$$
Is this square still a pullback iff $f\perp g$? What if $\mathsf C$ is a topos (not well-pointed in general)?


I ask because definition 8.10 of [these SDG notes](http://www.fuw.edu.pl/~kostecki/sdg.pdf) by Kostecki seemingly define formal étaleness by using such an internal analogue of orthogonality.


Also, in the case $\mathsf C$ is a topos, can anything be said about the merely existence or uniqueness (without existence) of diagonal fillers in terms of the induced map $[B,X]\to [A,X]\times \_{[A,Y]}[B,Y]$ being respectively epic or monic? This would shed light on formal unramifiedness/smoothness for me.


**Added.** A similar definition of orthogonality (unique lifting) is 5.1 [here](https://books.google.co.il/books?id=kX0qpFU3DS4C&pg=PA148&lpg=PA148&dq=lifting+property+in+a+topos+pullback&source=bl&ots=zUykpSuuuE&sig=DkclSM-pdX5STLXmzID_iH-bLQs&hl=en&sa=X&redir_esc=y#v=onepage&q=lifting%20property%20in%20a%20topos%20pullback&f=false).



Internal characterizations of lifting properties?
The second square being a pullback is a strictly stronger condition than $f\perp g$. Mapping out of the unit object shows that it implies $f\perp g$; mapping out of other objects says that the strong condition is equivalent to $(W\otimes f) \perp g$ for all objects $W$ (or $W\times f$ in the cartesian-closed case; I'm not sure if you meant to restrict to that one).


By mapping out of the unit object, we see that if the induced map $[B,X] \to [A,X]\times\_{[A,Y]} [B,Y]$ is monic, then fillers are unique if they exist. And if this map is *split* epic, or more generally if it belongs to a class of maps with respect to which the unit object is projective, then fillers always exist. But in general in a non-well-pointed topos this map could be epic without any filler existing globally; epimorphy of this map says only that fillers exist "locally" in the sense determined by the topology of the topos.


