# Projective tensor product on projective LCM is exact

I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. I understood how to prove that complex

$$0 \leftarrow O \left( V \right) \stackrel{\pi}{\leftarrow} O \left( V \times V \right) \stackrel{d}{\leftarrow} O \left( V \times V \right) \leftarrow 0 \, \, \, \left( \mathcal{V} \right)$$ splits, where $$O \left( V \right)$$ is a space of holomorphic functions, $$\pi: g (x, y) \mapsto g(x, x)$$ and $$d: g (x, y) \mapsto (y-x) g(x, y)$$. I understood the isomorphism of complexes $$Y \hat{\bigoplus}_{O \left( V \right) } \left( \mathcal{V} \right)$$ and $$\left( \mathcal{U} \right)$$, but I couldn't understand why this complex $$Y \hat{\bigoplus}_{O \left( V \right) } \left( \mathcal{V} \right)$$ splits (preceding lemma was about splitting of complex $$\left( \mathcal{V} \right)$$). It seems that the additivity of the functor is not enough to preserve the splitting.