$g \colon X \to Y$ that satisfies the following conditions:

1.  $g_{|A} = g_0$ ($g$ extends $g_0$); and

2.  $p\circ g$ is ${{\mathcal{F}_{}}}$-close to $f$.

We have the following commutative diagram. $$\xymatrix@M=8pt@C=35pt{
    Y \ar[r]^{p} & Z  \\
    A \ar@{^{(}->}[r] \ar[u]_{g_0} & X \ar[u]^{f} \ar@{.>}[ul]_{g} \\
    }$$ Let $\mathcal{P} = p^{-1}({{\mathcal{F}_{}}})$ and ${{\mathcal{G}_{}}} = f^{-1}({{\mathcal{F}_{}}})$. Let $C \colon {{\mathcal{G}_{}}} \to \mathcal{P}$ be defined by the formula $C(f^{-1}(F)) = p^{-1}(F)$. Since $p$ is surjective, $C$ is a carrier. For each $F \in {{\mathcal{F}_{}}}$ and each $x \in f^{-1}(F) \cap A$ we have $g_0(x) \in p^{-1}(f(x)) \in p^{-1}(F)$; hence $g_0$ is carried by $C$.

Let $\mathcal{H}$ be a closed locally finite locally finite-dimensional cover of $X$ such that $\mathcal{H}$ refines ${{\mathcal{G}_{}}}$. If ${{\mathcal{F}_{}}}$ is an open cover, then $\mathcal{H}$ exists as a refinement of an open cover ${{\mathcal{G}_{}}}$. If ${{\mathcal{F}_{}}}$ is closed locally finite locally finite-dimensional, then we can take $\mathcal{H} = {{\mathcal{G}_{}}}$, which has the required properties by Lemma \[lem:pullback properties\]. Let $D \colon \mathcal{H} \to {{\mathcal{G}_{}}}$ be any map such that $H \subset D(H)$ for each $H \in \mathcal{H}$.

Observe that $C \circ D \colon \mathcal{H} \to \mathcal{P}$ is a carrier, $g_0$ is carried by $C \circ D$, $\mathcal{P}$ is an 