In Sec. 7 (p. 60) of Conley's $\textit{Isolated Invariant Sets and the Morse Index (1976)}$, the following passage appears:
(In fact any isolated invariant set in $S$ is isolated in $\Phi$. It seems there ought to be some way to say the index of $S'$ in $S$ is "greater than" its index in $\Phi$: obviously if the index in $S$ is $\bar{0}$ the index in $\Phi$ is also. Since there is more room to move around in $\Phi$, the latter index cannot imply as much about $S'$ as the former. Note that if $S'$ is isolated in $S$ and $\Phi$ is the product of $S$ with the flow in $\mathbb{R}^1$ defined by $dx/dt=x$, then the index of $S'$ in $\Phi$ is the suspension of its index in $S$; thus the suspension of $X$ should be "less than" $X$. So far as this author knows such questions have not been studied in this context.)
I'm a little confused generally about this passage, but I have been struggling specifically to understand the bold portion. Since Conley says it is obvious, it probably is, but not yet to me. My question: Why is the bold “obvious” sentence true?
Notes on the passage. $S'$ is an isolated invariant subset of the isolated invariant set $S$. $\Phi$ is what Conley calls a "local flow" which for him is roughly a state space together with a local flow map $\varphi^t$ defined for small times $t$, although often Conley simply refers to the state space alone by $\Phi$. I believe the indices referred to above are the homotopy Conley indices $h(S',S)$ and $h(S',\Phi)$ (Conley's notation) rather than the indices $I(S',S)$ and $I(S',\Phi)$ as connected simple systems (p. 51 of the text). For a definition of isolated invariant sets and the homotopy Conley index, I refer here.