Stable ∞-categories are additive

Proof. We show that (1) is equivalent to (2); the equivalence between (1) and (3) is dual. Consider the following commutative square:

Since this square is both a pushout square as well as a pullback square and isomorphisms are closed under pushouts and pullbacks, we see that \(f\) is an isomorphism if and only if the map \(0 \to \cofib (f)\) is an isomorphism, proving the claim. □