The Stacks project

Proof. Let $M = (X, p, m)$ be an object of $M_ k$. Then $M$ is a summand of $(X, 0, m) = h(X)(m)$. By Homology, Lemma 12.17.3 it suffices to show that $h(X)(m) = h(X) \otimes \mathbf{1}(m)$ has a dual. By construction $\mathbf{1}(-m)$ is a left dual of $\mathbf{1}(m)$. Hence it suffices to show that $h(X)$ has a left dual, see Categories, Lemma 4.43.8. Let $X = \coprod X_ i$ be the decomposition of $X$ into irreducible components. Then $h(X) = \bigoplus h(X_ i)$ and it suffices to show that $h(X_ i)$ has a left dual, see Homology, Lemma 12.17.2. This follows from Lemma 45.4.9. $\square$