Let $\mathscr E$ be a smooth topos, where I am using the terminology of nLab. In particular that imples the existence of a line object $R$ and an "infinitesimally thickened point", which is an object of $\mathscr E$ defined in the internal language of the topos as $D = \{x\in R\mid x^2=0\}.$ The exponential object $$ TX = X^D $$ is called the tangent space to $X$. There is also a notion of a tangent category $T\mathscr C$ for any category $\mathscr C,$ the objects of which pairs $(X,A)$ for $X$ objects in $\mathscr C$ and $A$ abelian groups in the overcategory $\mathscr C/X.$ Now we could study an object $X$ in $\mathscr E$ by studying the overcategory $\mathscr E/X,$ thus arises the following

Question: What is the relationship between the tangent space $TX$ and the tangent category $T(\mathscr E/X)$ for an object $X$ of $\mathscr E$?

These different tangent notions seem to me to correspond to the gross vs. petit topos point of view, which are not always equivalent. If the question is easier to answer in some standard model for synthetic differential geometry instead of an arbitrary smooth topos, I will be happy with that as well.


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