$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{D}$ be the inclusion of a full subcategory. Consider a functor $$F:\mathcal{C}^{op}\rightarrow \Ab.$$ I've often seen examples where this functor might not be representable, but represented by a functor in the larger category, i.e. that there exists a $D\in \mathcal{D}$ such that $$F\cong \Hom_{\mathcal{D}}(-,D).$$ This is of course the idea behind say Schlessinger's criterion, which tells us when a deformation functor is pro-representable, or it is also the philosophy of stacks (extend the category of schemes with stacks to get representable functors).

My question is if there exists a nice set of criteria to study the representabilitly of $F$ in $\mathcal{D}$? A naïve approach I had was to Kan extend the functor to $\mathcal{D}$ and then to study the representability of the Kan extension. However this seems to be a bad idea in general.

A concrete situation I've encountered this is the following: consider the full subcategory $\Ho(\Sm)\subset \Hotc$ of smooth manifolds in the homotopy category of pointed CW-complexes. In the larger category, checking whether a functor is representable is "easy" by Brown representability. Consider for instance $$F:\Ho(\Sm)^{\text{op}}\rightarrow \Ab,\quad X\mapsto H^n(X,F)$$ the singular cohomology of degree $n$. In $\Hotc$, it is represented by the $n$-th Eilenberg-Maclane space, which is a pointed CW-complex but not a smooth manifold. Without using the fact that the singular cohomology on $\Ho(\Sm)$ is the restriction functor of singular cohomology on topological spaces, can we show that it is representable?

This illustrates also why it is not a good idea to study this via Kan extension, as the Kan extension of singular cohomology does not agree with singular cohomology.