We know the sheaf of relative differential $\Omega^1_{X/Y}$ has a nice universal property so we can define it via that property.
Can we do the same for the sheaf of differential $1$-form $\Omega^1_M$ over the smooth manifold $(M,\mathcal{O}_M)$ where $\mathcal{O}_M$ is the sheaf of smooth functions over $M$?
Say I want $\Omega^1_M$ to be universal in the sense that there exists a certain type of new derivation $\mathop{\mathrm{nDer}}_{\mathbb{K}}(\mathcal{O}_M,\mathcal{F})$, s.t. $$\operatorname{\mathrm{Hom}}_{\mathcal{O}_M}(\Omega^1_M,\mathcal{F})\to \operatorname{\mathrm{nDer}}_{\mathbb{K}}(\mathcal{O}_M,\mathcal{F}),\alpha\mapsto \alpha \circ d$$ is an isomorphism, this isomorphism mimics this stacks project page
We may restrict on the category that we pick $\mathcal{F}$ from. And let the new form of derivation to be the old derivation with more rules, e.g. continuity providing the target is a sheaf of topological modules, commutes with infinite sum providing convergence. Also see this discussion Kahler differentials and Ordinary Differentials in mathoverflow.
The following pictures about the definition of $\Omega^1_M$ may help.
They come from the book Manifolds, sheaves, and cohomology by Wedhorn, Torsten.
Here premanifolds are manifolds without the condition that the underlying topological space is Hausdorff and second-countable. And $\mathcal{T}_M$ (I don't know which font did the author use but this is a T) is the sheaf associated to the tangent bundle over $M$.
