Clarify about monoidal categories

Question

If $\mathbf C$ is a category with finite coproducts (I'll indicate the coproduct with $+$) and initial object $I$, we can say that $(\mathbf C,+, I)$ is a (symmetric) monoidal category. Since a left adjoint preserves colimits, if $\mathbf C$ was closed the functor $(-)+ X$ would preserve colimits, in particular the initial object. However $I+X\cong X$, that in general is not isomorphic to $I$. Is this a way of proving that a monoidal category with the cocartesian structure can't be closed or am I missing something? Thank you

Answer 1

Yes, that's correct. You've shown that any cocartesian closed category is equivalent to the terminal category. So if you have a cocartesian category (monoidal category with the coproduct) that isn't equivalent to the terminal category, you know already that it can't be closed with respect to that monoidal structure.