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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 $f$?

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    @NoahSchweber I see. That seems however like the kind of notation that would require an explanation. – 2017-02-05
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    I guess my question would be whether or not it can be considered standard? – 2017-02-05
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    I've turned my comments into an answer, so hopefully the responses to it/votes on it will indicate whether this is indeed standard. – 2017-02-05
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    If you're looking for a name, I would call it the pushforward via $f$. You could also call it the initial structure with respect to $f$ (but anyone not familiar with category theory would not understand), or induced via $f$. I think pushforward is the standard terminology. So is "induced structure", but it's more ambiguous, I think. – 2017-02-06
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    @tomasz thank you. I was about to ask if [pushforward](https://en.wikipedia.org/wiki/Pushforward) would be reasonable :) – 2017-02-06

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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 $f$."

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    Thanks for this. What about in topology, where $X$ can be used ambiguously to refer to both set $X$ and top. space $(X,\tau)$. Would it still seem reasonable to write space $f(X)$ or $f((X,\tau))$? – 2017-02-05
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    @Anguepa To me it would, yes. I'd be slightly happier with $f(X, \tau)$, since it's very reasonable to consider different topologies on the same set, but if I were in a context where only one topology was being discussed I'd be happy to write $f(X)$. Actually, though, in my own writing I use calligraphic letters to denote sets equipped with any kind of structure - so I'd refer to the space $(X, \tau)$ as $\mathcal{X}$, similarly to a structure, and I'd write "$f(\mathcal{X})$" for the space so pushed forward. – 2017-02-05