Suppose $f:S\to T$ and $g:T\to S$ are injective. Define a function $\varphi\colon P(S)\to P(S)$ by $\varphi(X)=S\setminus g(T\setminus f(X))$.

If $X\subseteq Y\subseteq S$, then $S\setminus g(T\setminus f(X))\subseteq S\setminus g(T\setminus f(Y))$, and so $\varphi$ is monotone. Since $P(S)$ is a complete lattice, we may apply the Tarski-Knaster theorem to conclude that the set of fixed points of $\varphi$ is a complete lattice and thus nonempty.

Let $C$ be a fixed point of $\varphi$. We have

Hence $g|_{{T\setminus f(C)}}\colon T\setminus f(C)\to S\setminus C$ and $f|_{C}:C\to f(C)$ are bijections. We can therefore construct the desired bijection $h\colon S\to T$ by defining