  We present photoionization modeling of galaxy populations at $z\sim0$, $2$,
and $> 6$ to bridge optical and far-infrared (FIR) emission-line diagrams. We
collect galaxies with measurements of optical and/or FIR ([O III] 88
$\mathrm{\mu m}$ and [C II] 158 $\mathrm{\mu m}$) emission line fluxes and plot
them on the [O III]$\lambda5007/\mathrm{H\beta}$--[N
II]$\lambda6585/\mathrm{H\alpha}$ (BPT) and L([O III]88)/SFR--L([C II]158)/SFR
diagrams, where SFR is the star-formation rate and L([O III]88) and L([C
II]158) are the FIR line luminosities. We aim to explain the galaxy
distributions on the two diagrams with photoionization models that employ three
nebular parameters: the ionization parameter $U$, hydrogen density
$n_\text{H}$, and gaseous metallicity $Z_\text{gas}$. Our models successfully
reproduce the nebular parameters of local galaxies, and then predict the
distributions of the $z\sim0$, $2$, and $> 6$ galaxies on the diagrams. The
predicted distributions illustrate the redshift evolution on all the diagrams;
e.g., [O III]$/\mathrm{H\beta}$ and [O III]88/[C II]158 ratios continuously
decrease from $z > 6$ to $0$. Specifically, the $z > 6$ galaxies exhibit
$\sim\!0.5$ dex higher $U$ than low-redshift galaxies at a given $Z_\text{gas}$
and show predicted flat distributions on the BPT diagram at $\log{\mathrm{[O
III]/H\beta}} = 0.5$-$0.8$. We find that some of the $z > 6$ galaxies exhibit
high L([O III]88)/SFR ratios. To explain these high ratios, our photoionization
models require a low stellar-to-gaseous metallicity ratio or bursty/increasing
star-formation history at $z > 6$. The James Webb Space Telescope will test the
predictions and scenarios for the $z > 6$ galaxies proposed by our
photoionization modeling.
