Short exact sequence of persistence modules

Question

I am currently trying to work out a elementary proof of the following statement: Let $X$ be a simplicial complex with a filtration $\mathbb{X}: X=\bigcup_{n\in\mathbb{N}} X_n$, let $k\in\mathbb{N}$ and let $K$ be a field. Then: \begin{align} BC_k(\mathbb{X};K)_\infty \overset{f}{=} BC_{k}(X,\mathbb{X};K)_\infty,\end{align} where the second equality is given by the identification via the bijection $[a,\infty)\overset{f}{\rightleftarrows}(-\infty,a]$. Here BC denotes the corresponding barcode of the persitence module over the filtration. As a reference I am using notation from https://arxiv.org/pdf/1107.5665.pdf and the proof outline from the introduction part of https://arxiv.org/pdf/2012.12881.pdf. I know that it says that in later sections this argument will be provided in more detail, but I try to do this more direct as I only need it for the statement above. I assume that the following statement as of Bauer 2021 holds: \begin{align}H_d(\mathbb{X};K)_\infty\cong im( \eta_d)\rightleftarrows im(\epsilon_d)\cong H_d(X,\mathbb{X};K)_{\infty}. \end{align} With the long exact sequence of pairs I get the following short exact sequence. $\require{AMScd}$ \begin{CD} 0@>>>im( \eta_k)@>{f}>>& H_{k}(X;K)@>{g}>>im (\epsilon_{k})@>>>0\end{CD} Because f is injective I know that death indices must map to death indices in the corresponding barcodes. And for g surjective births are mapped to births. This seemed to solve the issue but I realized that the correspondance that I get now is from $\infty$ to $-\infty$ since $(a,\infty)\in BC(im(\eta_k))$ and $(-\infty,b)\in BC(im(\epsilon_k))$. But I need the correspondence that the birth indices in $BC(im(\eta_k))$ are the death indices in $BC(im(\epsilon_k))$, i.e., $a=b$ but I got that the death indices are the birth indices. What am I missing?