Proposition 5.3.25. For a semiadditive \(\infty \)-category \(C\), the hom functor \(\Hom _C\colon C\catop \times C \to \An \) lifts uniquely to a functor \[ \Hom _C\colon C\catop \times C \to \CMon (\An ) \] which preserves direct sums in both variables. If \(C\) is additive, this lands in \(\CGrp (\An )\).
Proof. This follows directly from the universal property of \(\CMon (\An )\) established in Proposition 5.3.23; the proof is completely analogous to that of Proposition 4.4.4 and is omitted. □
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