 Solution

No such set of propositions exists. Suppose that $S$ had precisely the total injections as its models, fix $x_0\in X$, and adjointo $S$. Every finite subset of the enlarged theory has a model. Indeed, it excludes only finitely many possible values $F\subset Y$ for $x_0$. Starting with any injection $h:X\to Y$, either $h(x_0)\notin F$, or one may choose $x_1\ne x_0$ with $h(x_1)\notin F$ and swap the values of $x_0$ and $x_1$.

The propositional compactness theorem would therefore give a model of the whole enlarged theory. Its relation $f_v$ is a model of $S$ but has no value at $x_0$, contradicting the assumed description of the models of $S$. This is the compactness obstruction to expressing totality over an infinite codomain.