Recognizing exact squares

Exercise 4.2.19 (Recognizing exact squares). Let \(C\) be a stable \(\infty \)-category and consider a commutative square in \(C\):

The following conditions are equivalent:

(1)

The square is exact;

(2)

The sequence \(X \xrightarrow {(f,u)} Y \oplus Z \xrightarrow {{v \choose -g}} W\) is an exact sequence.

Hint. Compare the original square with the exactness of the displayed sequence by means of the diagonal \(W\to W\oplus W\) and the difference map \(W\oplus W\to W\). The necessary comparison square is exact because its associated shear map is an isomorphism.