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?
The specific situation this came up in was trying to prove an equivalence of categories between sheaves on two different sites. I didn't want to deal with showing things were sheaves so I wanted to just prove an equivalence between the presheaf categories and say we can sheafify to get an equivalence on the sheaf level. I ended up proving that my functor sent sheaves to sheaves directly but I was still wondering if there was an answer to this question in general.
