Let $\varphi:\mathscr{F}\to\mathscr{G}$ be a morphism of presheaves over a topological space $X$ with values in an abelian category $\sf{C}$. As usual, we define the image presheaf $\operatorname{im}\varphi$ as being $\operatorname{im}\varphi(U):=\operatorname{im}\varphi_U$, where $\operatorname{im}\varphi_U$ is the kernel of the cokernel of $\varphi_U$. Surely, since limits and colimits in functor categories are computed pointwise, that coincides with the kernel of the cokernel of $\varphi$.
I know the restriction maps of both the kernel and the cokernel presheaves, which defines the restriction maps of the image presheaf. Nevertheless, I think that maybe we could have a simpler description of those maps by using the universal property of the image.
Let $V\subset U$ be a pair of nested open sets in $X$ and consider the commutative diagram below.
I think that maybe we could use the universal property of $\operatorname{im}\varphi_V$ to obtain our map $\operatorname{im}\varphi_V\to\operatorname{im}\varphi_V$. For that, it suffices to find a monomorphism $\operatorname{im}\varphi_V\to \mathscr{G}(U)$ but I fail to see what that would be. I would appreciate if anyone knows how to define such a monomorphism or if anyone know a better way of constructing these restriction maps.
Just for being sure that we're on the same page, the universal property of the image says that the image of $\varphi:A\to B$ is a monomorphism $i:K\to B$ such that $\varphi$ factors through $i$ and that is initial with these properties. That is, if $L\to B$ is another monomorphism through which $\varphi$ also factors, then it exists a unique morphism $K\to L$ such that the diagram
commutes.