Suppose I have 2 filtrations of simplicial complexes $K,G$: $\{K_{\alpha}\}_{\alpha\in\mathbb{R}},\{G_{\beta}\}_{\beta\in\mathbb{R}}$. Here, $K_{\alpha}$ is a subcomplex of $K$ and if $\alpha\leq\beta$ then $K_{\alpha}\subseteq K_{\beta}$.
Because of this filtration, we have the following persistence module of homology groups:
$$ i:H_d(K_{\alpha})\rightarrow H_d(K_{\beta}) $$ where $i$ is the homology map induced by the inclusion map $K_{\alpha}\subseteq K_{\beta}$. We have a similar persistence module for the filtration $\{G_{\beta}\}_{\beta\in\mathbb{R}}$.
Now, suppose we have a simplicial map $f:K\rightarrow G$ such that $f(K_{\alpha})\subseteq G_{\alpha+\epsilon}$ for all $\alpha\in\mathbb{R}$. In the following paper, (https://arxiv.org/pdf/1207.3885.pdf, Pg 4, Ex 2.2), and others I have read, this simplicial map induces chain maps between $H_d(K_{\alpha})$ and $H^{\epsilon}_d(G_{\alpha})$. $H^{\epsilon}_d(G_{\alpha})$ is just the chain of modules $H_d(G_{\alpha})$ shifted down by $\epsilon$.
In other words, for all $\alpha\in\mathbb{R}$ there is a $\phi_{\alpha}:H_d(K_{\alpha})\rightarrow H_d(G_{\alpha+\epsilon})$ that would satisfy the usual commutative properties after shifting the chain.
However, I don't see how such a map $\phi_{\alpha}$ can be defined. Firstly, if we have a n-simplex $[v_0,...,v_n]$, $[f(v_0),...,f(v_n)]$ need not be a $n$-simplex. Secondly, if $y\in\ker(\partial_{\alpha}:C_{d}(K_{\alpha})\rightarrow C_{d-1}(K_{\alpha}))$ how do we know that $f(y)\in \ker(\partial_{\alpha}:C_{d}(G_{\alpha+\epsilon})\rightarrow C_{d-1}(G_{\alpha+\epsilon}))$.
Does someone know how such a chain map arises?