Suppose we have $C_{n-1} \subseteq C_{n}$ and $A \subseteq B$ a morphism $f \colon B \to C_n$ in Set such that we have a bijection induced by $f$.
$f\colon B -A \to C_n - C_{n-1}$.
It follows that $C_n$ is the pushout of the diagram $B\setminus A \leftarrow \emptyset \to C_{n}$. Is it true that $C_n$ is the pushout of the diagram $B \leftarrow A \to C_{n-1}$. I am working in the category of set. I think it is true as we have this bijection and given any cocone $P$ and morphism $\alpha \colon C_{n-1} \to P$ and $\beta \colon B \to P$ then I can define $F \colon C_n \to P$ as $F(x) = \alpha(x)$ if $x\in C_{n-1}$ and if $x \notin C_{n-1}$ then by the bijection I have that $ x$ belongs to $B$ and therefore $F(x) = \beta(f^{-1}(x))$ does the job. I don't understand why it would be universal and where the fact that $B \setminus A$ is in bijection is used. If I replace $B \setminus A$ by $B$ it would also have worked, I guess.