A question on an answer on Math Overflow about the tangent bundle I have a question on the accepted answer of this Math Overflow question.
Let $K$ be a field and $X$ a $K$-scheme. Define the morphism of schemes $T=\operatorname{Spec}\operatorname{Sym}(\Omega_{X/K}) \to X$ as the tangent bundle of $X$ as suggested in the answer. As it is said there, this is not always locally trivial.
Further it is said (somehow) in the answer that $T(C)=\operatorname{Hom}_{\operatorname{Spec}(K)}(\operatorname{Spec}(C[\varepsilon]/(\varepsilon^2)),X)$ for a $K$-algebra $C$ which I don't understand. How can you insert $C$ into a scheme $T$? Should this be $$T(Z)=\operatorname{Hom}_{\operatorname{Spec}(K)}(\operatorname{Spec}(C[\varepsilon]/(\varepsilon^2)),X)$$ for an open and affine subscheme $Z=\operatorname{Spec}(C)$ of $X$ instead?
Is it (theorefore?) possible to understand the ''whole bundle'' $T$ as follows: There is the notion of a hom-sheaf
$$
V\mapsto \mathcal{H}om_{\operatorname{Spec}(K)}(\operatorname{Spec}(K[\varepsilon]/(\varepsilon^2))|_V,X|_V)=:S(V)
$$
which is a functor $S:(Schemes/K)^{op}\to Sets$.

Is this functor representable by the scheme $T$, i.e. is there an isomorphism
  $$ \operatorname{Hom}_{\operatorname{Spec}(K)}(-,T) \cong S(-)$$
  of functors $(Schemes/K)^{op}\to Sets$?

 A: Regarding your first question, the answer is no, you do not need to assume that $Z = \operatorname{Spec}(C)$ is an open affine subscheme of $X.$ The notation $T(C)$ refers to the $C$-valued points of $X,$ where $C$ is, as mentioned, an arbitrary $K$-algebra. In other words, $T(C) = \operatorname{Hom}_K(\operatorname{Spec}(C),T),$ by definition.
Your confusion probably arises because the equality mentioned, $$T(C) = \operatorname{Hom}_K(\operatorname{Spec}(C[t]/t^2),X)$$ is actually a specification of what we want the $C$-valued points of $T$ to be, before actually knowing $T$ exists. That is, there should be, and is, an isomorphism of functors from $T(-)$ to $\operatorname{Hom}_K(\operatorname{Spec}(-\times_K K[t]/t^2),X).$ This is what it means for the functor on the right hand side to be representable (by the scheme $T$).
Actually, we have to be a bit more careful, since the domain of our functors here is actually only affine schemes. To fully prove representability, we need to show that we can glue the local data. There is a nice description of the construction of the tangent bundles and jet schemes (the higher order version) in some papers of Mustata-Ein and Ishii on the arXiv.
