Given a function $f: B \rightarrow C$ and a bijection $g: B' \rightarrow B$, there is a naturally induced function $f': B' \rightarrow C$, namely the composition $f' = f \circ g$.
Now, given a function $f: A \times B \rightarrow C$ and a bijection $g: B' \rightarrow B$, there is also a naturally induced function $f': A \times B' \rightarrow C$ defined by $f'(a, b') = f(a, g(b'))$. Is there some standard notation for this function in terms of $f$ and $g$?
Of course, one could define a new function $g': A \times B' \rightarrow A \times B$ by $g'(a, b') = (a, g(b))$. Then $f'$ really is just $f \circ g'$ as in the previous case. But this is clunky and I was hoping there was an elegant notation from algebra or elsewhere analogous to $\circ$ in the one dimensional case.
You could write $f \circ (\mathrm{id} \times g)$.