It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (say, over $\mathbb C$), the category of coherent sheaves of $\mathcal O_X$-modules is abelian.
Now, a rigid analytic space $X$ inherits a Grothendieck topology and we can similarly define coherent sheaves of $\mathcal O_X$-modules with respect to this. So I would like to ask
Question: Do coherent sheaves on rigid analytic spaces form an abelian category?
In the first place, we know from BGR's book that taking kernels and images will preserve coherence in this setting. So the answer might be positive. But the axioms of abelian categories are too abstract for me, and I have no idea about how to prove or disprove this.