at (180:0.58) [$y_1$]{}; at (250:0.58) [$x_1$]{}; at (-20:0.58) [$y_0$]{}; at (150:0.58) [$z_1$]{}; (60:0.48) – (60:0.52); (70:0.48) – (70:0.52); (50:0.48) – (50:0.52); (180:0.48) – (180:0.52); (190:0.48) – (190:0.52); (170:0.48) – (170:0.52); (240:0.48) – (240:0.52); (260:0.48) – (260:0.52); (250:0.48) – (250:0.52); (-20:0.48) – (-20:0.52); (-10:0.48) – (-10:0.52); (-30:0.48) – (-30:0.52); (140:0.48) – (140:0.52); (150:0.48) – (150:0.52); (160:0.48) – (160:0.52); (60:0.5) .. controls (60:0.3) and (250:0.3) .. (250:0.5); (180:0.5) .. controls (180:0.3) and (-20:0.3) .. (-20:0.5); (60:0.5) .. controls (60:0.3) and (150:0.3) .. (150:0.5);

Precovering subcategories of the cluster categories ${\mathscr{C}}({\mathscr{Z}})$ {#sec:precovering}
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This section provides the following main ingredient for the proof of Theorem \[thm:A\].

\[T:precovering\] Let ${\mathscr{X}}$ be a set of diagonals of ${\mathscr{Z}}$. Then $\add E({\mathscr{X}})$ is a precovering subcategory of ${\mathscr{C}}({\mathscr{Z}})$ if and only if ${\mathscr{X}}$ satisfies conditions PC1 and PC2 from Definition \[def:PC\].

The proof can be found at the end of the section. First we require some preparation, not least the following definition due to [@E sec. 1].

\[Precovers\] \[D:precovering\] Let ${\mathscr{T}}$ be a category, $X\subseteq {\mathscr{T}}$ a full subcategory.

1.  Let $t\in {\mathscr{T}}$ be an object. An object $x\in X$ together with a morphism $f \colon x \to t$ is called an [*$X$-precover of $t$*]{} if 