Let $\mathcal{C}'$ and $\mathcal{D}'$ be categories so that $\mathcal{C} \subseteq \mathcal{C}'$ and $\mathcal{D} \subseteq \mathcal{D}'$ are full subcategories. Suppose the forgetful functors $F_{\mathcal{C}}:\mathcal{C} \to \mathcal{C}'$ and $F_{\mathcal{D}}: \mathcal{D} \to \mathcal{D}'$ have left adjoints $G_{\mathcal{C}}$ and $G_{\mathcal{D}}$. Then in general does an equivalence of categories between $\mathcal{C}'$ and $\mathcal{D}'$ induce an equivalence of categories between $\mathcal{C}$ and $\mathcal{D}$ by composing with the adjunctions? If this is not true in general are there any criteria to guarantee this?
I think that in this generality the answer is "no". For example, take $\mathcal{C}'=\mathcal{D}'$ equal to some additive category, take the identity as the equivalence, take $\mathcal{C}=\mathcal{C}'$ and $\mathcal{D}=0$. This seems to satisfy your hypotheses.