$\newcommand{\op}{\mathsf{op}}\newcommand{\yo}{よ}$Given a functor $F\colon\mathcal{C}\to\mathcal{D}$, the composition
$$\mathcal{D}\overset{\yo}{\hookrightarrow}\mathsf{PSh}(\mathcal{D})\xrightarrow{F^*}\mathsf{PSh}(\mathcal{C})$$
is called the **restricted Yoneda embedding** associated to $F$. Write $\yo_F$ for it.

I'm interested in understanding what commonly imposed conditions on functors, such as essential injectivity, fullness, faithfulness, etc., would imply for $F$ when imposed on $\yo_F$, as well as characterisations of functors $F$ such that $\yo_F$ satisfies a certain condition.

For example, we have the following result for fully faithfulness:

Proposition.The restricted Yoneda embedding $\yo_F$ is fully faithful precisely when $F$ is dense.

Qiaochu Yuan's blog post here has the following result for faithfulness when $F$ is a full subcategory inclusion:

Proposition.The following conditions are equivalent:

- The restricted Yoneda embedding $\yo_F\colon\mathcal{D}\to\mathsf{PSh}(\mathcal{C})$ is faithful.
- The functors $\mathrm{Hom}_{\mathcal{C}}(F(A),-)$ are jointly faithful.
- (If $\mathcal{C}$ has coproducts) For each $D\in\mathrm{Obj}(\mathcal{D})$, the map $$\coprod_{\substack{A\in\mathrm{Obj}(\mathcal{C})\\f\colon F(A)\to D}}F(A)\to D$$ is an epimorphism.

Are there characterisations for when $\yo_F$ is:

- Faithful;
- Full;
- Faithful on isomorphisms;
- Full on isomorphisms;
- Pseudomonic;
- Conservative;
- Essentially injective?

Moreover, when $\yo_F$ satisfies one of these conditions, what can we deduce for $F$, even if the implication only goes in one direction?