Well-definedness of the Fitting ideal ( Mostly Commutative Algebra, Theorem (3.9.6). )

I am reading the Antoine Chambert-Loir, (Mostly) Commutative Algebra, Theorem 3.9.6 and stuck at understanding some statement. Let $A$ be a commutative ring. For every matrix $U\in \mathrm{Mat}_{n,m}(A)$ and every positive integer $p$, let us denote by $\Delta_p(U)$ the ideal of $A$ generated by the determinants of all $p \times p$ matrices extracted from $U$ ( a minor of size $p$ of $U$ ) . We also write this ideal as $\Delta_p(u_1, \dots, u_m)$, where $u_1, \dots, u_m \in A^n$ are the columns of $U$. Let $M$ be a finitely generated $A$-module and let $\varphi : A^n \to M$ be a surjective morphism of $A$-modules. For every integer $p\in \{ 0, \dots, n\}$, let then $J_p(\varphi)$ be the ideal of $A$ generated by ideals $\Delta_p(U) = \Delta_p(u_1, \dots, u_p)$, where $U= ( u_1 ,\dots, u_p)$ ranges over all subsets of $\mathrm{Ker}(\varphi)^p$ ( the direct product of $\mathrm{Ker}(\varphi)$ ). One has $J_0(\varphi)=A$. If $p<0$, then we set $J_p(\varphi) = A$. Lemma (3.9.5). Let $p$ be an integer such that $1\le p\le n$. (a) $J_p(\varphi) \subseteq J_{p-1}(\varphi)$ (b) Let $(v_i)_{i \in I}$ be a family of elements of $\mathrm{Ker}(\varphi)$ which generates $\mathrm{Ker}(\varphi)$. Then $J_{p}(\varphi)$ is generated by the elements $\Delta_p(v_{i_1}, \dots, v_{i_p} )$, where $i_{1}, \dots, i_{p} \in I$ (c) $\mathrm{Ann}_A(M) \cdot J_{p-1}(\varphi) \subseteq J_p(\varphi)$. Theorem (3.9.6) ( Well-definedness of Fitting ideal ). Let $A$ be a commutative ring, let $M$ b ea finitely generated $A$-module. Let $\varphi : A^m \to M$ and $\psi : A^{m} \to M$ be surjective morphisms. For every integer $d \ge 0$, the ideals $J_{m-d}(\varphi) $ and $J_{n-d}(\varphi)$ are equal. Errata: In the second paragraph of the step 2) , $\theta ''(x,y) := \psi(y)$ is more correct. Can anyone explain the underlined statement more friendly? Q. Why all generators of $J_{p-1}(\psi)$ can be obtained as minors of size $p$ of a suitable matrix $Z$?