Existence of (strong epi,mono)-factorization in finitely well-complete category

Question

I am reading Borceux’s Handbook of categorical algebra, volume 1. The proposition 4.4.3 proves that finitely well-complete category has (strong epi,mono)-factorization, but I don’t understand the proof.

First, for a morphism $f : A \to B$, he takes every possible factorization $f=i_kp_k$ with $i_k$ a monomorphism. My first question is that if this factorization always exists.

Assuming at least one such pair exists, he computes the intersection of such $i_k$s, say $i : I \to B$. Then he says there is $p:A\to I$ such that $f=ip$. But I don’t know how to take such $p$. My another question is if we can take such $p$. If $I$ is some sort of limit, we can take $p$ by universal property of the limit, however, since the class of the equivalence classes of $i_k$s may not be finite or even a small set, I don’t think this would work.

I would welcome any help!

Answer 1

First, for a morphism $f:A→B$, he takes every possible factorization $f=i_kp_k$ with $i_k$ a monomorphism. My first question is that if this factorization always exists.

You have $f = 1\circ f$ as a possible factorization!

Assuming at least one such pair exists, he computes the intersection of such $i_k$s, say $i:I→B$. Then he says there is $p:A→I$ such that $f=ip$. But I don’t know how to take such $p$.

The intersection is defined as the product of the subobjects. Since you have projections $p_k$, there's a unique $p$ that factors through all of them. (Using the universal property of the product.)

The rest of the proof is showing this $p$ is indeed strong-epi.

however, since the class of the equivalence classes of $i_k$s may not be finite or even a small set, I don’t think this would work.

That's what the "finitely well-complete" assumption is for.