A question about the Yoneda embedding

Question

Write $y:\mathcal{C}\to[\mathcal{C}^{\text{op}},\mathbf{Set}]$ for the Yoneda embedding on a small category $\mathcal{C}$.

If there is a category $\mathcal{D}$ and functors $G:\mathcal{C}\to\mathcal{D}$ and $F:\mathcal{D}\to[\mathcal{C}^{\text{op}},\mathbf{Set}]$ such that $y=F\circ G$, is it necessary that $\mathcal{D}$ becomes a subcategory of $[\mathcal{C}^{\text{op}},\mathbf{Set}]$?

Answer 1

No, just adjoin a disjoint object to $C$ with however many endomorphisms, and factor Yoneda through the resulting $D$ by sending the new object, say, to the terminal preheaf.

Or better yet, let $D$ be the disjoint union of two copies of the presheaf category.