Partial maps are representable in the category of presheaves on C

Question

I know $Set^{C^{op}}$ is a topos and in toposes partial maps are representable but indeed I want to construct the partial map classifier $\tilde{F}$ for a presheaf F and a mono $\eta:F\longrightarrow \tilde{F}$ which classifies partial maps to F. I know how to construct (find) $\tilde{F}$ with Yoneda lemma (For an object c in C, $\tilde{F}(c)$ can be the set of all compatible families in F indexed by a sieve R on c) and actually I know what $\eta$ is but I want to know how I can find or construct that $\eta$ without knowing what it is (maybe with Yoneda lemma).