To the point:
What does being isomorphic over/under an object $c$ mean?
Let $x,y,c$ be objects in a category $\mathsf C$, and let's concentrate on "over". I think "$x\cong y$ over $c$" should mean that there is an isomophism $h:f\to g$ in the slice category $\mathsf C/c$ between some morphisms $f:x\to c$ and $g:y\to c$. This of course implies that $x\cong y$ in $C$. but is stronger. Graphically, there should exist $f,g$ and $h$ ($h$ an iso in $\mathsf C$) such that this diagram commutes:
$\hspace{4.25cm}$
But could it somehow mean that for all $f:x\to c$ and all $g:y\to c$, we have $f=gh$? This sounds too strong to me, but it would remedy the fact that $f$ and $g$ should be considered part of the data in my first interpretation.
For context, I read the claim in a exercise by Riehl that if $F$ and $G$ are naturally isomorphic functors $\mathsf C\to \mathsf{Set}$, then "$\int\! F \cong \int\! G$ over $\mathsf C$", where $\int\!F$ is the category of elements. Here, the relevant slice category is $\mathsf{CAT}/\mathsf C$, where $\mathsf{CAT}$ is the category of locally small categories. I've found a couple of other uses of the same phrasing through google, some by Riehl as well.
I interpret the claim as pertaining specifically to the canonical projections to $\mathsf C$. I.e. for a natural isomorphism $\alpha:F\Rightarrow G$, we have an isomorphism of categories $\int\!\alpha:\int\!F\to\int\!G$ (my notation), such that this diagram of functors commutes:
$\hspace{4.25cm}$
Here, $\int\!\alpha: (c,x) \mapsto (c,\alpha_cx), f \mapsto f$.
This should all be true, but is it the right interpretation of the claim?