Kernel and cokernel of sheaves of rings over a topological space

Question

Let $\phi:F\to G$ be a morphism of presheaves of rings over a topological space $X$. One can define $Ker(\phi)$ and $Coker(\phi)$.

Since $F,G$ are presheaves of rings, for each open $U\subset X$, $Ker(\phi)(U)$ is not a unital ring but rather a module of $F(U)$. For $Coker(\phi)(U)$, I have it as a quotient module of $G(U)$ which is a module over $F(U)$ at best. They are not rings?

What is the meaning of $Ker(\phi),Coker(\phi)$ in the case of morphism of presheave of rings here? If they are presheaves of abelian category objects, it would make sense for some time.

Answer 1

As stated in the comments, the target category is not the usual one of unital rings, but instead the category of (possibly) nonunital rings; i.e. associative $\mathbb{Z}$-algebras.

This category has a zero object, so the (co)kernel of a map is defined as the (co)equalizer of that map with the parallel zero map.

Limits of presheaves are given pointwise, so $\ker(\phi)(U)$ is the kernel of $F(U) \to G(U)$, and similarly for the cokernel.