For a functor $F: \mathcal{C^{\mathrm{op}}} \to \mathrm{Set}$ and an object $A \in \mathcal{C}$, we have $\mathrm{Nat}(\mathcal{C}(-, A), F) \cong FA$, which is the Yoneda Lemma. (There is a co-Yoneda Lemma concerning $\mathrm{Nat}(G, \mathcal{C}(A, -))$ as well, where $G: \mathcal{C} \to \mathrm{Set}$.)
What can we say about $\mathrm{Nat}(F, \mathcal{C}(-, A))$?
If you set $\mathcal{D} = \mathcal{C}^{op}$ (this is just a notational aid) then $F : \mathcal{D} \to \mathsf{Set}$ is a covariant functor, and a natural transformation $F \to \mathcal{C}(-,A)$^is the data of morphisms $F(X) \to \mathcal{C}(X,A) = \mathcal{D}(A,X)$ satisfying a naturality condition, which is exactly the same whether you consider $F$ as covariant or contravariant. In other words $\operatorname{Nat}(F, \mathcal{C}(-,A)) = \operatorname{Nat}(F, \mathcal{D}(A,-))$. Thus this is just an application of the co-Yoneda lemma that you mentioned, except you apply it to $F$ viewed as a functor $\mathcal{D} \to \mathsf{Set}$.