Beck torsors, formally unramified objects, and K\"ahler differentials

Let $\mathcal{C}$ be a category with pullbacks. We define a $\textit{Beck torsor}$in$$\mathcal{C}$$asamorphism$Z\to Y$in$$\mathcal{C}$$ which is a torsor for a Beck module over $Y$. We say that an object $X$ of $\mathcal{C}$ is $\mathit{\text{formally}}\mathit{\text{unramified}}$ if, for every Beck torsor $Z\to Y$ in $\mathcal{C}$, the canonical map $$\text{Hom}${$\mathcal{C}$}(X, Z)\to $\text{Hom}${$\mathcal{C}$}(X, Y)$isinjective\mathrm{.}If$A$ is a commutative ring with identity, then an $A$-algebra $B$ is formally unramified in the category of $A$-algebras if and only if the ring homomorphism $A\to B$ is formally unramified. Given that $A\to B$ is formally unramified if and only if ${\mathrm{\Omega }}_{B\mathrm{/}A}=0$, we seek a similar classification for general formally unramified objects. We say that $\mathcal{C}$ has $\textit{K"ahler differentials}$if,foreachobject$X$of$$\mathcal{C}$$,theforgetfulfunctor$$\text{Ab}$($\mathcal{C}$/X)\to $\mathcal{C}$/X$ from the category of Beck modules over $X$ has a left adjoint $\Omega: $\mathcal{C}$/X\to $\text{Ab}$($\mathcal{C}$/X)$\mathrm{.}Ourmainresultisthatif$$\mathcal{C}$$ has K"ahler differentials, then an object $X$ of $\mathcal{C}$ is formally unramified if and only if ${\mathrm{\Omega }}_X$ is a zero object in $\text{Ab}(\mathcal{C}\mathrm{/}X)$.