Module of differentials in the functorial approach to schemes and quasi-coherent modules Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules $M_A$ for $A$-points $s \in X(A)$. The compatibility is given by (coherent) isomorphisms $M_A \otimes_A B \cong M_B$ for homomorphisms $A \to B$ (inducing a  $B$-point of $X$). In other words,
$$\mathsf{Qcoh}(X) = \int_{A} \mathsf{Mod}(A)^{X(A)}.$$
In this functorial approach to quasi-coherent modules, I wonder how to define the module of differentials $\Omega^1_{X/R}$ (in such a way that we recover the usual module in the case that $X$ is a scheme)? Notice that $\Omega^1_{X/R}|_A = \Omega^1_{A/R}$ won't be correct; this even fails when $X$ is a scheme.
So probably one should first describe the $A$-module $\Omega^1_{X/R} |_A$ if $X$ is a scheme and $\mathrm{Spec}(A) \to X$ is a morphism of schemes. When $\mathrm{Spec}(A) \to X$ is étale, then the result is $\Omega^1_{A/R}$, but otherwise it will be something more complicated. I've tried to use the fundamental sequences for $\Omega^1$, but didn't succeed. Perhaps it's not even possible to describe $\Omega^1_{X/R} |_A$? Are things better in derived algebraic geometry?
 A: Let $\mathbf{Qcoh}_R$ be the total category of modules:


*

*The objects are pairs $(A, M)$ where $A$ is a commutative $R$-algebra and $M$ is an $A$-module.

*The morphisms $(A, M) \to (B, N)$ are pairs $(f, g)$ where $f : A \to B$ is an $R$-algebra homomorphism and $g : M \to N$ is an abelian group homomorphism such that $g (a m) = f (a) g (m)$.

*Composition and identities are the obvious ones.


Then there is an evident projection $\mathbf{Qcoh}_R \to \mathbf{CAlg}_R$ and it is a Grothendieck opfibration. As you say, a quasicoherent module over a functor $X : \mathbf{CAlg}_R \to \mathbf{Set}$ is simply a cocartesian functor $\mathbf{El} (X) \to \mathbf{Qcoh}_R$ over $\mathbf{CAlg}_R$, where $\mathbf{El} (X)$ is the category of elements of $X$, i.e. the comma category $(1 \downarrow X)$.
Now, the assignment $A \mapsto \Omega^1_{A \mid R}$ defines an $\mathscr{O}_R$-module, i.e. a section $\Omega^1_R : \mathbf{CAlg}_R \to \mathbf{Qcoh}_R$ of the Grothendieck opfibration $\mathbf{Qcoh}_R \to \mathbf{CAlg}_R$, and as you observed, it is not a quasicoherent module, i.e. it is not a cocartesian section. Nonetheless, it is a good starting point: after all, for any $R$-scheme $X$ and any affine open subscheme $U \subseteq X$, $\Gamma (U, \Omega^1_R)$ is naturally isomorphic to the $\mathscr{O}_X (U)$-module $\Gamma (U, \Omega^1_{X \mid R})$. 
Accordingly, let $\Omega^1_X : \mathbf{El} (X) \to \mathbf{Qcoh}_R$ be the composite
$$\require{AMScd}
\begin{CD}
\mathbf{El} (X) @>>> \mathbf{CAlg}_R @>{\Omega^1_R}>> \mathbf{Qcoh}_R
\end{CD}$$
and let $\mathbf{Mod}_R (X)$ be the category of all (not necessarily cocartesian) functors $\mathbf{El} (X) \to \mathbf{Qcoh}_R$ over $\mathbf{CAlg}_R$. There is an evident full inclusion
$$\mathbf{Qcoh}_R (X) \hookrightarrow \mathbf{Mod}_R (X)$$
and I believe what we want is a coreflection for $\Omega^1_X$, i.e. a quasicoherent module $\tilde{\Omega}^1_X$ and a homomorphism $\tilde{\Omega}^1_X \to \Omega^1_X$ such that, for every object $M$ in $\mathbf{Qcoh}_R (X)$, every homomorphism $M \to \Omega^1_X$ factors through $\tilde{\Omega}^1_X \to \Omega^1_X$ uniquely.
If I'm not mistaken, $\mathbf{Qcoh}_R (X)$ and $\mathbf{Mod}_R (X)$ are always cocomplete with colimits computed componentwise, so the inclusion preserves colimits. Moreover, when $X$ is nice enough (e.g. a scheme, but perhaps more generally a colimit of a small diagram of representable functors $\mathbf{CAlg}_R \to \mathbf{Set}$ where the arrows are flat morphisms), $\mathbf{Qcoh}_R (X)$ is a Grothendieck abelian category, and I think the same is true for  $\mathbf{Mod}_R (X)$. (This should just be the fact that the 2-category of locally presentable abelian categories and exact left adjoints is closed under pseudolimits and lax limits.) Thus the inclusion has a right adjoint and we can construct $\tilde{\Omega}^1_X$.
Everything I wrote here should be taken with a pinch of salt. I have only verified it in the case where $X$ is representable (i.e. an affine $R$-scheme) – but that's all quite trivial.
