For the sake of generality I phrase this question in a model theoretic context, although ideally I am asking about a term that is of common usage in other areas of mathematics, and in particular in topology and order theory.
Let $\mathcal{X}=(X,\phi_1,\phi_2,\ldots)$ be an $\mathcal{L}$-structure in the model-theoretic sense and let $f:X\rightarrow Y$ be a bijection.
Is there a standard notation for the $\mathcal{L}$-structure $\mathcal{Y}=(Y,\phi_1,\phi_2,\ldots)$ that is isomorphic to $\mathcal{X}$ through $f$?
E.g. Structure induced by $f$? Transfer through $f$? Space isomorphic to $\mathcal{X}$ through $f$?
I believe the standard notation is just "$f(\mathcal{X})$", which is an abuse of notation, but a pretty benign one: if $f:X\rightarrow Y$, what else could that refer to?
This also reflects other benign abuses of notation - e.g. writing "$x\in\mathcal{X}$" instead of "$x\in X$".
(I've also heard $\mathcal{Y}$ described as "$\mathcal{X}$ pushed through $f$," but much more rarely - and I actually find it less clear than "$f(\mathcal{X})$.")
That said, it certainly never hurts to explain notation: if you want to be absolutely certain, you could say at the beginning of the paper/book "We write "$f(\mathcal{X})$" for the structure on $Y$ induced by $f$."