$\textbf{Motivation for Question:}$ If there is some graph $G = (V, E)$ which is isomorphic to some other graph $G' = (V', E')$, for any function, $f$ defined on $G$, which takes some element(s) of $G$ and outputs an element(s) of $G$ (essentially, any function on $G$), when $f$ is applied to $G'$ would it necessarily produce the corresponding value in $G'$, just as an example $ \phi : v \mapsto v', v \in V, v' \in V'$ where $\phi$ represents the function which returns the vertex in $V'$ that $v$ is "renamed" as in the isomorphic graph $G'$, then $\phi(f(v)) = f(v')$. I can see that it works for something like $\delta_G (v)$ or $\text{Bellman-Ford}$. However, is this true in general?
$\textbf{Problem Statement:}$ There are two sets $ X, Y$ and that there is an $f$ such that $X \cong Y$. Consider an arbitrary function $f_1$ and elements $x_1, x_3 \in X, x_2 \in Y$ such that $f_1 (x_1) = x_3, f(x_1) = x_2$. Is $f(f_1(x_1)) = f_1(x_2)$?
$\textbf{Background:}$ I'm not experienced in Algebra, I am taking a fundamental course on Groups and Rings, this is a question that came up when I was talking about isomorphisms between rings.
Note: I did read this answer: Is there a property that is not preserved by isomorphism?. However, I think my question is a little more nuanced since I'm not really talking about equivalences (there was another question about it that I read which cleared that up).
