Please correct me if I get the language (or math) wrong below, I'm relatively uneducated in category theory and homological algebra.
I'm reading about sheaves in Hartshorne's book, and he mentions that a presheaf can be regarded as a contravariant functor from $\mathscr{T}(X)$ to $\mathfrak{A}\mathfrak{b}$, where $\mathscr{T}(X)$ is the poset category of open sets on $X$, and $\mathfrak{A}\mathfrak{b}$ is the category of abelian groups. When he then defines a morphism of presheaves, the definition seems to be equivalent to saying that a morphism $\phi : P_1 \to P_2$ is a natural transformation between $P_1$ and $P_2$ regarded as functors.
Later, Hartshorne defines the kernel, cokernel, and image of a presheaf, and neglects to include a proof that each of these is again a presheaf. I can check that directly from the definitions he gives, but I'm wondering whether there's an easier way to see this using homological algebra/category theory? For example, if I have a natural transformation whose target is an abelian category, can I define a "kernel functor" taking each object $U$ to the kernel of the component $\eta_U$, and each morphism to some kind of induced morphism? Then I could see clearly that the kernel, cokernel and image of a morphism of presheaves is again a presheaf, since these would each just be compositions of functors.
Best Answer
The general fact is that in any functor category $[C, D]$, if a limit or colimit of some shape exists in $D$ then it exists in the functor category and it is computed pointwise. What this means, explicitly, is that if $F : J \to [C, D]$ is a diagram of shape $J$, then
$$(\lim_j F(j))(c) = \lim_j \left( F(j)(c) \right)$$
and the same for colimits.
So in particular if $D$ is the category of abelian groups (or another abelian category) then it has kernels and cokernels and so does the functor category and they are computed pointwise. This is a good exercise and can be used to show that if $D$ is an abelian category then so is the functor category $[C, D]$, hence in particular so is the category of $D$-valued presheaves.
It's worth noting as a warning that the same is still true for limits of sheaves but not for colimits of sheaves; these are not computed pointwise and in general one has to sheafify.