Proposition 16.4.5 (Path components). The adjunction
is symmetric monoidal on the left and lax symmetric monoidal on the right, where \(\Ab \) is equipped with its usual tensor product.
Proof. The functor \(\pi _0\colon \An \to \Set \) is a symmetric monoidal left adjoint for the cartesian monoidal structures and preserves finite products. Thus Proposition 16.4.2 applies. Finally, Example 16.4.4 identifies the symmetric monoidal structure on \(\CGrp (\Set )=\Ab \) with the usual tensor product. □
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