Karoubianness of a triangulated category

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New enhancements of derived categories of coherent sheaves and applications (2014, unknown): “orphism. Recall the enhancement C+,b Coh(InjSh(X×X)) of Db(Coh(X×X)) −→∼ Db Coh(Sh(X×X)) from section 3.1. Note that T ∈C +,b Coh(InjSh(X×X)) by Theorem 2.5.(b) and that Db(Coh(X ×X)) is Karoubian by [LC07]. We are allowed to apply Proposition B.1.(a) in this setting and see that F := HomC(Sh(X×X))(”
Symmetric products of mixed Hodge modules (2010, methodology): “rem (1.9), and get the canonical action of Sn on π∗⊠ nM ∈ DbMHM(SnX) which is compatible with the one on the underlying Q-complexes. By the splitting of an idempotent in DbMHM(SnX) (see [BS] and also [LC] for a simpler argument) which is applied to e1 in (2.1.1), we get a direct factor SnM of π∗⊠nM ”
Parity Sheaves (2009, background): “By our assumptions on k, D(X) is a Krull-Schmidt category (see [LC07]) and hence the endomorphism ring of an indecomposable object in D(X) is local.”
Quiver algebras as Fukaya categories (2013, background): “agrees with its own idempotent completion, see [31].”