In the book Manifolds, Sheaves, and Cohomology by Wedhorn appears the following equality.
$$\mathrm T_pX= \left\{ \substack{t:\mathrm{pt}[\varepsilon]\to M\text{ morphism of locally} \\\text{ringed spaces with }t(\ast)=p} \right\} $$ Here $X$ is a differentiable premanifold viewed as a locally ringed space (structure sheaf given by differentiable real functions), and $\mathrm{pt}[\varepsilon]$ is the locally ringed space given by the singleton with structure sheaf given by the ring of dual numbers $\mathbb R[\varepsilon]$.
What is the bijection here?
Trying to unpack the RHS, the sheaf morphism $t^\ast \mathrm{D}_X(V)\to \mathbb R[\varepsilon]$ of $t$ itself seemingly amounts to a local ring homomorphism $$\mathrm{D}_{X,p}\longrightarrow \mathbb R[\varepsilon]$$ from the stalk at $p$ of the structure sheaf $\mathrm{D}_X$ of $X$ to the ring of dual numbers. How are such things in bijection with, say, geometric tangent vectors (derivatives of differentiable curves in $X$ based at $p$)? The former deals with real functions from the premanifold while the latter deals with curves into the premanifold. (The only such local homomorphism that appears natural to me is $f\mapsto f(p)+f^\prime(p)\varepsilon$, and I don't know what to do with it anyway.)
Let $A$ be a commutative $\mathbb{R}$-algebra. A ring homomorphism $\varphi : A \to \mathbb{R}[\epsilon]$ can be alternatively described as:
The relationship is given by the formula $$ \varphi(a) = \theta(a) + d(a) \epsilon $$
In the case that $A$ is an algebra of functions and $\theta$ is evaluation at the point $x$, the above identity has the form $$ d(fg) = f(x) d(g) + g(x) d(f) $$ This is precisely the usual characterization of tangent vectors as derivations.
E.g. given a "geometric" tangent vector $v$ to the point $p$, we can construct such a homomorphism $D_{X,p} \to \mathbb{R}[\epsilon]$ by the formula
$$ f \mapsto f(p) + \nabla_v f(p) \epsilon $$