$\require{AMScd}$Let $F:\mathcal{C}^{op}\times \mathcal{C} \to \mathcal{D}$ be a functor and $\mathcal{C}' \subseteq \mathcal{C}$ a full subcategory. Assume that the coends $C$ over $F$ and $C'$ over $F|_{\mathcal{C}'}$ exist. By the universal property of $C'$, we obtain a map $i:C' \to C$.

I am looking for a necessary and sufficient condition for this to be a monomorphism.

As it is pointed out in the comments, **the following part doesn't really make sense**:

I think we can consider $C$ as a pushout of the diagram $$ \begin{CD} \coprod_{c\in\mathcal C'} F(c,c) @>>> C' \\ @VVV \\ \coprod_{c\in\mathcal C} F(c,c) \end{CD} $$ in the category of cocones $Cocone(F)$ of $F$. Hence, if coprojections are monos in $\mathcal{D}$ and pushouts preserve monos in $Cocone(F)$, $i$ is a monomorphism. However, the last condition seems to be a bit strong, so is there another argument I'm not seeing? In particular, is it true for abelian categories?