Isomorphic kernels imply pullback?

Question

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma:

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He implies (but doesn't say) that the same is not true if we replace equality with isomorphism. But I can't see why it wouldn't be. Do you agree, or have I missed something?

Answer 1

The full version of the lemma is treated on page 92 of Zimmermann's Representation Theory: A Homological Algebra Approach. It is interesting to note that he uses only the relevant limit properties, so the result holds in any category with a zero object, provided that the relevant kernels exist.

Answer 2

You are right that the result is also true if you replace the equality by an isomorphism. The reason it is called "a partial converse" is that there is the extra assumption that $\nu'$ is an epimorphism; you don't need that to prove that a pullback square induces an isomorphism between the kernels (which is what Lemma 1.2 claims). This hypothesis is not (explicitly) present in Zimmerman, but that's a mistake : at the end of the proof he uses the short five lemma, but without the assumption that $\alpha$ is an epimorphism, the short sequences are not exact.

By the way, it is not true that the result holds in any category with a zero object. It is true for one direction (the one given in Lemma 1.2. (i)), but the other direction requires (and is in fact equivalent to) the short five lemma, so it doesn't hold in the category of pointed sets, or in the category of monoids.