You have two functions $f: A \rightarrow A,\quad g: B \rightarrow B$, which map onto their own domain, and a bijective function $t: A \rightarrow B$, which transforms back and forth between the domains of $f$ and $g$. They have the relation $\forall a\in A: t(f(a)) = g\big(t(a)\big)$.
Is there a name for this thing or for something more general, like when $f$ and $g$ are not required to map onto their own domains?
Else, can we transform this into something with a known name? This is something related to an isomorphism, but an isomorphism requires a binary relation, not an unary operation/function $t$.
Update 2022-05-30:
I think I was wrong when I said the above, which is now strike-through text. I think an isomorphism doesn't require a binary relation. Is this correct?
Then this thing is simply an isomorphism with an unary operation.
