Consider the situation described in the following diagram, namely:
- $A$, $A'$, $B$, and $B'$ are sets.
- $\alpha:A\rightarrow A'$ and $\beta:B\rightarrow B'$ are bijections.
- $f:A\rightarrow B$ and $\ f':A'\rightarrow B'$.
- The following equations are satisfied. $$ f = \beta^{-1} \circ f' \circ \alpha\\ f' = \beta\circ f \circ \alpha^{-1} $$
In a sense $f$ and $f'$ are the same function, in that each can be computed in terms of the other, with no information being lost or gained.
Is there an accepted terminology for this sameness of $f$ and $f'$?
You could say that $f$ and $f'$ are isomorphic, since the functions $\alpha$ and $\beta$ define an isomorphism from $f$ to $f'$ in the arrow category $\mathbf{Set}^{\to}$.
Here's some details:
Given a category $\mathcal{C}$, the arrow category $\mathcal{C}^{\to}$ has the morphisms of $\mathcal{C}$ as its objects and commutative squares in $\mathcal{C}$ as its morphisms.
That is, a morphism from $(f : A \to B)$ to $(f' : A' \to B')$ is a pair $(\alpha,\beta)$ consisting of a morphism $\alpha : A \to A'$ and a morphism $\beta : B \to B'$, such that $\beta \circ f = f' \circ \alpha$. An isomorphism in $\mathcal{C}^{\to}$ is simply a pair $(\alpha,\beta)$ of isomorphisms in $\mathcal{C}$.
When $\mathcal{C} = \mathbf{Set}$, this says that an isomorphism from a function $f : A \to B$ to a function $f' : A' \to B'$ in $\mathbf{Set}^{\to}$ is a pair of bijections $\alpha : A \to A'$ and $\beta : B \to B'$ such that $\beta \circ f = f' \circ \alpha$—this is exactly your situation.