  Let $\left(X_n, d_n\right)$ be a sequence of metric spaces and let
$\mathcal{F}=\left\{f_n\right\}_{n \in \mathbb{Z}}$ be a sequence of continuous
and onto maps $f_n: X_n \rightarrow X_{n+1}, n \in \mathbb{Z}_{+}$. In this
paper, we prove that if the compression ratio meets $\prod \lambda_i=0$, then
there exists $\delta_n>0$ such that any $\delta_n$ - pseudo-orbit
$\left\{x_n\right\}_{n \in \mathbb{Z}}$ is $\varepsilon$ - shadowed by a unique
point $x \in X_0$. For the asymptotic average shadowing property, we prove that
if $\left.\mathcal{F}\right|_A$ has asymptotically average shadowing property,
then $\mathcal{F}$ also has a asymptotically average shadowing propertywhen a
density-related condition is satisfied. Additionally, the conclusion that the
shadowing performance of strong equicontinuity and pseudo-shadowing property
implies limit shadowing is also obtained. Furthermore, the shadowing property
of non-autonomous product space is also discussed.
