Full subcategory of abelian category is abelian

Question

I'm trying to understand a proof in Rotman's 'Introduction to Homological Algebra', Proposition 5.92, p.310.

Proposition: Let $\mathcal S$ be a full subcategory of an abelian category $\mathcal A$. If, for all $A, B \in \text{obj}(\mathcal S)$ and all $f : A \to B$,

1. a zero object in $\mathcal A$ lies in $\mathcal S$,
2. the direct sum $A \oplus B$ in $\mathcal A$ lies in $\mathcal S$,
3. both $\text{ker } f$ and $\text{coker } f$ lie in $\mathcal S$,

then $\mathcal S$ is an abelian category.

The definition for abelian category that is being used: A category $\mathcal C$ is an abelian category if it is an additive category such that

1. every morphism has a kernel and cokernel
2. every monomorphism is a kernel and every epimorphism is a cokernel.

Here's the proof given in the book:

The hypothesis gives $\mathcal S$ additive, so $\mathcal S$ is abelian if axiom 2 in the definition of abelian category holds. If $f : A \to B$ is a monomorphism in $\mathcal S$, then $\text{ker } f = 0$. But $\text{ker } f$ is the same in $\mathcal A$ as in $\mathcal S$, by hypothesis, so that $f$ is monic in $\mathcal A$. By hypothesis, $\text{coker } f$ is a morphism in $\mathcal S$. As $\mathcal A$ is abelian, there is a morphism $g : B \to C$ with $f = \text{ker } g$. But $g$ is a morphism in $\mathcal S$, because $\mathcal S$ contains cokernels, and so $f = \text{ker } g$ in $\mathcal S$.

I understand that $\text{coker } f$ is a morphism in $\mathcal S$.

I also understand that since $\mathcal A$ is abelian, there exists a morphism $g : B\to C$ with $f = \text{ker } g$ (this is the second axiom of the definition of abelian category being applied to $\mathcal A$).

I don't understand how these two pieces are connected, but from the way he's using them, it appears that he's assuming $g = \text{coker } f$. I don't see how this follows, since we're only given that there exists a function $g$ in $\mathcal A$ with $f = \text{ker } g$.

The main piece of the proof I don't understand is the line "But $g$ is a morphism in $\mathcal S$, because $\mathcal S$ contains cokernels".

I understand $\mathcal S$ contains cokernels, but I don't understand why that implies that $g$ is in $\mathcal S$, unless $g$ is a cokernel of a morphism in $\mathcal S$. My guess was that $g = \text{coker } f$, but I'm unable to show this. (I've tried using the definition of cokernel as the solution to the universal mapping problem to show that $g = \text{coker }f$, specifically showing domain and codomain are the same, or equivalent, but it hasn't gotten me anywhere).

Also, as a side question: the first axiom in the proposition statement seems out of place. I think that should be before: "If, for all $A, B$...". Correct?

Answer 1

It simply doesn't follow that $g$ is a morphism in $S$. For example, $A$ might be the category of abelian groups, $S$ might be the full subcategory of finitely generated abelian groups, and the target of $g$ might be an infinitely generated abelian group.

But the proof is very easy to repair: just define $g = \text{coker}(f) \in A$ in the first place. Then in fact we have $g \in S$, so $\text{ker}(g) \in S$ as well. Now, a key feature of full subcategories $S$ is that any limits or colimits, in $A$, of $S$-valued diagrams which happen to land in $S$ must in fact be limits or colimits in $S$: that is, inclusions of full subcategories reflect limits and colimits. Because $f = \text{ker}(g)$ in $A$, it follows that $f = \text{ker}(g)$ in $S$ as well, so $f$ is a kernel. Similarly for epimorphisms.