Exactness on $\sf{QCoh}_X$ versus on $\mathscr{O}_X$-$\sf{Mod}$. Let $X$ be a scheme and $\sf{QCoh}_X$, $\sf{Coh}_X$, $\mathscr{O}_X$-$\sf{Mod}$ be the categories of quasi-coherent sheaves, coherent sheaves and $\mathscr{O}_X$-modules over $X$, respectively. All these categories are Abelian, which makes me wonder the following:

*

*If $\varphi:\mathscr{F}\to\mathscr{G}$ is a morphism of coherent sheaves, the kernel and the of $\varphi$ in those 3 categories coincide? In other words, a sequence of coherent sheaves is exact in $\sf{Coh}_X$ iff it is in $\sf{QCoh}_X$ and iff it is in $\mathscr{O}_X$-$\sf{Mod}$?

 A: Let $\sf{B}$ be an Abelian category and $\sf{A} \subset \sf{B}$ be an additive subcategory in the sense that $\hom_\sf{A}(M,N) \cong \hom_\sf{B}(M,N)$ as Abelian groups. Let $\phi:M \to N$ be a morphism in $\sf{B}$ and consider its kernel $K$, that is, we have an exact sequence
$$
0 \to K \to M \to N
$$
Suppose now that $M, N$ and $K$ lie in the (essential) image of $\sf{A}$. Then $K$ is the kernel of $M \to N$ seen as a morphism in $\sf{A}$:
By our assumption the composite $K \to M \to N$ is still $0$, and given any map $T \to M$ in $\sf{A}$ with $T \to M \to N$ equal to the zero map then we can find a unique lift $T \to K$ in $\sf{B}$, but since $\sf{A}$ is full, this is a unique lift in $\sf{A}$ as well.
By using a dual argument we conclude that the same is true for cokernels.
The upshot then is that if we have a full subcategory of an Abelian category that is closed under Kernel and cokernel (at least in the essential image) then the inclusion functor is exact, that is, this is an Abelian subcategory.
Now we just note that this is true for quasi-coherent and for coherent modules.
