Let $C$ be a category. For the purposes of this question, I would like to avoid cases where the answer might be "no" simply because $C$ is "too large", and so I will ask that $C$ has a set of **generators**, i.e. a set of objects $\{X_\alpha\}_{\alpha \in A}$ such that for any two morphisms $f,g : Y \to Z$ that are not equal, there exists an $\alpha\in A$ such that $\hom(X_\alpha,f) \neq \hom(X_\alpha,g)$ as maps $\hom(X_\alpha,Y) \to \hom(X_\alpha,Z)$.

(I am most interested in the specific cases where $C$ is either small or presentable. More generally, $C$ could be a "site" or even a "colimit sketch", and this question would still be apt.)

I would like to advocate for the following terminology: a **cosheaf** on $C$ is a covariant functor $F : C \to \mathrm{Set}$ that takes colimits to colimits. (Similarly, a **sheaf** is *contravariant* functor $C \to \mathrm{Set}$ that takes colimits to limits.)

Given two nonequal morphisms $f,g: Y \to Z$, does there necessarily exist a cosheaf $F$ such that $F(f) \neq F(g)$ as maps $F(Y) \to F(Z)$?

Note that the usual version of the Yoneda lemma doesn't suffice: the corepresentable functor $\hom(X,-)$ preserves limits, not colimits, and the representable cofunctor $\hom(-,X)$ is a sheaf, not a cosheaf. Using representable sheaves, one immediately has the answer "yes" if $\mathrm{Set}$ were replaced by $\mathrm{Set}^{\mathrm{op}}$. Perhaps there is a faithful cocontinuous functor $\mathrm{Set}^{\mathrm{op}} \to \mathrm{Set}$ that I am not aware of?

Some applications of cosheaves that I care about are listed at http://ncatlab.org/nlab/show/cosheaf; the proposition there also provides one reason I care about this question.