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 the following condition:

For each diagonal $Y = \{y_0,y_1\}$ of ${\mathscr{Z}}$ there is a finite set of diagonals $X^1, \ldots, X^l \in {\mathscr{X}}$, such that for each $X=\{x_0,x_1\} \in {\mathscr{X}}$ with $$x_0 \leqslant y_0 \leqslant x_1^{--} < x_1 \leqslant y_1 \leqslant x_0^{--}$$ there is an $i \in \{1, \ldots, l\}$ with $X^i = \{x^i_0,x^i_1\}$ and $$x_0 \leqslant x^i_0 \leqslant y_0 \text{ and } x_1 \leqslant x^i_1 \leqslant y_1.$$

This is immediate by combining Section \[sec:IT\](vii) with Lemma \[L:precover decomposition\].

\[P:precovering implies PC1 and PC2\] Let ${\mathscr{X}}$ be a set of diagonals of ${\mathscr{Z}}$. If $\add E({\mathscr{X}})$ is a precovering subcategory of $\cC( \cZ )$ then ${\mathscr{X}}$ satisfies conditions PC1 and PC2.

Let ${\mathscr{X}}$ be a set of diagonals such that $\add E({\mathscr{X}})$ is precovering. We show that ${\mathscr{X}}$ satisfies condition PC1. The fact that ${\mathscr{X}}$ satisfies condition PC2 follows by an analogous argument. Hence let $X^i = \{x^i_0, x^i_1\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ be a sequence from ${\mathscr{X}}$ with $x^i_0 \to p$ from below and $x^i_1 \to q$ from below with $p \neq q$.

If $p,q \in {\mathscr{Z}}$, we have 