In page 362 of tom Dieck's Algebraic Topology, the author gives a definition for the tangent space of a premanifold (locally ringed space locally isomorphic to open subset of Euclidean space) which I understand as follows:
Let $(X,\mathcal O _X)$ be a premanifold. A tangent space at $p$ consists of a pair $(\mathrm T_pX,\jmath)$ satisfying the following data.
- $\mathrm T_pX$ is a vector space.
- For each chart $\bf x$ of $X$ about $p$, $\jmath_\mathbf{x}:\mathrm T_pX\to \mathbb R ^n$ is a linear isomorphism.
- $\jmath_\mathbf{y}\circ \jmath_{\mathbf{x}}^{-1}=\mathrm d_{\mathbf x p}(\mathbf y\circ \mathbf x^{-1}):\mathbb R ^n \cong \mathbb R^n$.
These properties imply that if $(\mathrm T_pX,\jmath)$ and $(\mathrm T_p^\prime X,\jmath ^\prime)$ are two tangent spaces of $X$ at $p$ then the composite $\jmath_\mathbf{x}^{-1}\circ \jmath_\mathbf{x}^\prime:\mathrm T^\prime_pX\cong \mathrm T_pX$ does not depend on the chart $\mathbf x$. From this the author concludes the tangent space is unique up to unique isomorphism "by the universal property".
What are the categories involved and in what precise sense is the tangent space an initial/terminal object?
It seems that he's just proving that the category of tangent spaces is a contractible grouped: take the category of all tangent spaces at $p$ with morphisms the linear maps factoring through the $j$ maps for some chart. Then you've shown there exists a unique morphism between any two tangent spaces, and all morphisms are isomorphisms.