Addition of morphisms in a semiadditive ∞-category

Remark 4.2.16. For objects \(X,Y\) in a semiadditive \(\infty \)-category \(C\), we may equip \(\pi _0\Hom _C(X,Y)\) with the structure of an abelian monoid: given morphisms \(f,g\colon X \to Y\), we define their sum \(f + g\) as the following composite: \[ X \xrightarrow {(\id ,\id )} X \oplus X \xrightarrow {f \oplus g} Y \oplus Y \xrightarrow {{\id \choose \id }} Y. \] We leave it to the reader to check that this is indeed unital, associative and commutative. Then \(C\) is additive if and only if \(\pi _0\Hom _C(X,Y)\) is in fact a group. In particular, for every morphism \(f\colon X \to Y\) we may form its negative \(-f\colon X \to Y\) which satisfies \(f + (-f) = 0\).