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I am working with tensor products and there are a lot of identities, etc. that are true after appropriate identifications (between tensor products) are made.

For example, if $V$ is a vector space, then $V\otimes \mathbb{C}\cong V$. Suppose I have an element $\lambda \otimes v$. In my work I might want to write

$$\lambda\otimes v=\lambda v,$$

but instead have been writing

$$\lambda\otimes v\cong \lambda v.$$

How bad is this notation?

To answer this question I suppose I could just introduce the notation; day that for elements $a$ in a space $A$ and $b$ in a space $B$, $a\cong b$ means that there is an isomorphism $\Phi:A\rightarrow B$ and $\Phi(a)=b$... or would the notation should be clear from context (or even state that it should be clear from context what $a\cong b$ means?)?

Has anyone got a better notation for "equal via isomorphism"?

This question is very close to this.

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I think it's fine to use "$=$" so long as you put a small note specifying that this is an extension (hardly an "abuse") of the notation. For example:

When I use "$=$" between elements of two different objects I mean that there is a canonical isomorphism between those two objects and that these elements are equal after applying this isomorphism.

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I think that the cleanest way is to define this as an equivalence relation. Show that this is indeed an equivalence relation.

As notation, your notation or $\equiv$ are good.

In my opinion it would be bad to just use equal, there would most likely be some places where confusing the two different meanings of the $=$ sign would lead to wrong results.

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