nLab very unequivocally states that $FinVect_k$ (finite dimensional vector spaces over the field $k$) is a closed category. However, I have a fake proof below that $FinVect_k$ is not a closed category.
Question: Where are the mistakes in the fake proof that $FinVect_k$ is not a closed category?
Fake proof: Wikipedia says that a category is closed if and only if it possesses an internal Hom functor. E.g. for a category $C$ which is a subcategory of $Set$, both $Hom(\cdot, B)$ and $Hom(A, \cdot)$ are functors whose target category is $C$ itself, rather than only just $Set$.
One can observe that the "relation" on $C \times C$ which takes a $k$-vector space $V$ to its dual $V^*$ is not a functor. But by definition $V^* = Hom(V, k)$, so that a contravariant Hom functor is what takes a vector space to its dual, and since the relationship between $V$ and $V^*$ is not functorial in $FinVect_k$, the contravariant Hom functor for $FinVect_k$ must only be a functor when considered as a map into $Set$. (I.e. one can not relate $V$ and $V^*$ canonically/functorially using linear maps -- we only have a canonical/functorial relationship between $V$ and its double dual.)
Since the contravariant Hom functor $Hom(\cdot, k)$, taking a vector space to its dual, is not a functor $FinVect_k \to FinVect_k$, the contravariant Hom functor in general can not be considered a functor into $FinVect_k$, so $FinVect_k$ has no internal Hom, so it is not a closed category.
Your underlying error seems to be that you expect $\hom(\cdot, B)$ to be a functor $\mathcal{C} \to \mathcal{C}$, but that's wrong — it should be a functor $\mathcal{C}^\mathrm{op} \to \mathcal{C}$.