I know that in general, the association of a group to its own automorphism group is not functorial, c.f. this question. But what about vector spaces? Let’s fix a field $k$, and take two vector spaces $V$ and $V^\prime$ over $k$ as well as the set $\operatorname{Hom}_{\operatorname{Vect}}(V,V^\prime)$ of $k$-linear maps between them. There also exists a set of group homomorphisms $\operatorname{Hom}_{\operatorname{Grp}}(\operatorname{Aut}(V),\operatorname{Aut}(V^\prime))$. Now in order to lift the map $$\operatorname{Aut}(-):V\longmapsto\operatorname{Aut}(V)$$ to a functor from the category of vector spaces over $k$ to the category of groups, we would have to specify in some way how this map acts on morphisms (Its action on objects is quite clear). I don’t see a natural way to identify linear maps $\lambda:V\to V^\prime$ with group homomorphisms $\tilde{\lambda}:\operatorname{Aut}(V)\to\operatorname{Aut}(V^\prime)$. Simply defining $\operatorname{Aut}(-)$ to trivially satisfy the axioms of a functor by sending the identity map $V\to V$ to the identity homomorphism $\operatorname{Aut}(V)\to\operatorname{Aut}(V)$ and sending compositions of linear maps to compositions of group homomorphisms feels wrong, and there seems to be no guarantee that the map defined in such a way really exists. Is there a proper way to define such a functor?
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