each morphism $g \colon x' \to t$ with $x' \in X$ factors through $f$. That is, there exists a morphism $h \colon x' \to x$ such that $g = f\circ h$. $$\xymatrix{& x \ar[d]^f \\
              x' \ar[r]_-g \ar@{-->}[ru]^h & t}$$

2.  The subcategory $X \subseteq {\mathscr{T}}$ is called [*precovering*]{} if each object $t\in {\mathscr{T}}$ has an $X$-precover.

Let $\cT$ be an additive category. An [*additive subcategory*]{} $X$ of $\cT$ is a full subcategory of $\cT$ closed under isomorphisms, finite direct sums, and direct summands.

Since ${\mathscr{C}}({\mathscr{Z}})$ is Krull-Schmidt, its additive subcategories are determined by the indecomposable objects they contain. Thus, there is a one-to-one correspondence between additive subcategories of ${\mathscr{C}}({\mathscr{Z}})$ and sets of diagonals of ${\mathscr{Z}}$.

Given a set of diagonals ${\mathscr{X}}$ we write $E({\mathscr{X}})$ for the corresponding set of indecomposable objects of ${\mathscr{C}}({\mathscr{Z}})$. The corresponding additive subcategory of ${\mathscr{C}}({\mathscr{Z}})$ is given by $\add E({\mathscr{X}})$.

\[L:precover decomposition\] Let $D \subseteq {\mathscr{C}}({\mathscr{Z}})$ be an additive subcategory, $e \in {\mathscr{C}}({\mathscr{Z}})$ an indecomposable object, and $$\delta \colon d_1 \oplus \ldots \oplus d_n \to e$$ a morphism in ${\mathscr{C}}({\mathscr{Z}})$ with $d_i \in D$ indecomposable for each $i \in \{1, \ldots, n\}$.

We can write $\delta = (\delta_1, \ldots, \delta_n)$, and $\delta$ is a $D$-precover 