Infinite product of a short exact sequence

Question

I was trying to show that in an abelian category satisfying (AB4)* the product of a short exact sequence is a short exact sequence.

Given $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$

exact then the corresponding sequence is exact:

$0 \rightarrow\prod{A_i} \rightarrow \prod{B_i} \rightarrow \prod{C_i} \rightarrow 0$

I could show that the product of monomorphisms is a monomorphism and that the product of the kernels is the kernel of the product but I need to show that the image of the product is the product of the images.

I think this should be the same as the kernel part but I'm missing an arrow.

Answer 1

If you know that a product of epimorphisms is an epimorphism and product of monomorphisms is a monomorphism, then you know that products preserve images. Indeed, the image factorization of a map $f:A\to B$ is uniquely determined (up to canonical isomorphism) by the fact that it is a factorization $A\to I \to B$ of $f$ as a composition of an epimorphism $A\to I$ followed by a monomorphism $I\to B$.