I'm probably just being dense, but I'm stuck on the following. Let $\mathcal{E}$ be a category satisfying the Giraud axioms: namely $\mathcal{E}$

1. is locally presentable,
2. has universal colimits,
3. has disjoint coproducts, and
4. has effective equivalence relations.

Using only this, prove that all epimorphisms in $\mathcal{E}$ are effective.

My strategy so far has gone as follows. First, I found this way to prove all epimorphisms are effective given that I can prove that $\mathcal{E}$ has epi-mono factorization systems, has stable pullbacks and is balanced.

From Moerdijk-Maclane Page 577, we can construct a epi-mono factorization of any morphism $f : X \to Y$ in $\mathcal{E}$ using the regular co-image, i.e. the coequalizer of the kernel pair $ X \times_Y X \rightrightarrows X \to \text{coim}(f)$. Universal colimits implies that epimorphisms are stable under pullback (I think). But how do I prove that $\mathcal{E}$ is balanced? I was trying to use the epi-mono factorization system, but it seems to me that it is not unique in the sense that I need.

In other words, it seems that the Moerdijk-Maclane construction is only unique in the sense of an (effective epi, mono)-factorization system. If I can factor $f$ by an effective epi followed by a mono, then I can prove its iso to the coimage, but what if it is just factored as an epi followed by a mono? Then I can't seem to prove anything.

Any help would be appreciated.

I'm stuck on the following. Let $\mathcal{E}$ be a category satisfying the Giraud axioms: namely $\mathcal{E}$

1. is locally presentable,
2. has universal colimits,
3. has disjoint coproducts, and
4. has effective equivalence relations.

Using only this, prove that all epimorphisms in $\mathcal{E}$ are effective.

My strategy so far has gone as follows. First, I found this way to prove all epimorphisms are effective given that I can prove that $\mathcal{E}$ has epi-mono factorization systems, has stable pullbacks and is balanced.

From Moerdijk-Maclane Page 577, we can construct a epi-mono factorization of any morphism $f : X \to Y$ in $\mathcal{E}$ using the regular co-image, i.e. the coequalizer of the kernel pair $ X \times_Y X \rightrightarrows X \to \text{coim}(f)$. Universal colimits implies that epimorphisms are stable under pullback (I think). But how do I prove that $\mathcal{E}$ is balanced? I was trying to use the epi-mono factorization system, but it seems to me that it is not unique in the sense that I need.

In other words, it seems that the Moerdijk-Maclane construction is only unique in the sense of an (effective epi, mono)-factorization system. If I can factor $f$ by an effective epi followed by a mono, then I can prove its iso to the coimage, but what if it is just factored as an epi followed by a mono? Then I can't seem to prove anything.

Any help would be appreciated.