Let $(\mathcal{C},\otimes, I)$ be a symmetric monoidal closed category (with $\otimes$-unit $I$).
Suppose that $\mathcal{D}$ is a (full) subcategory of $\mathcal{C}$, which contains $I$, and which is closed under $\otimes$, i.e. $A,B\in\mathcal{D}$ implies $A\otimes B\in \mathcal{D}$.
Does it follow that $(\mathcal{D},\otimes)$ becomes a symmetric monoidal closed category itself?