This is the Lemma 9.11 of Rotman's "An Introduction to Algebraic Topology".
The topic where I found this simple lemma of homological algebra is the Theorem of Acyclic Models. So we are talking about a category with models.
First of all those are the Lemma and its proof:
This lemma is a fundamental key to the theorem of acyclic models. The advantage of this lemma is to complete the diagram with a natural transformation $\gamma$ using the exactness of the bottom row ONLY on the models $\mathcal{M}$ of the category. The proof uses the analog obvious result for abelian groups (or in general for projective objects).
Keeping in mind the reason for existing of this lemma, I don't understand why we don't take the hypothesis to the first row to be a complex ONLY in models $\mathcal{M}$ exactly as in the second row. In my opinion, the book proves this "stronger" result.
I'm asking this because I found the same Lemma with the same proof also in "Lectures on Algebraic Topology" by Albrecht Dold (11.5 Lemma). I really can't see where is the problem assuming the hypothesis of the first row to be a complex only on models. This seems to me like we want to prove something stronger then what we have but not the "strongest" we can do.