nLab Sandbox2

This can be illustrated using string diagrams, as follows. The companion and conjoint of $f$ are the horizontal arrows

equipped with 2-cells

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metric	Minkowski	Schwartzschild	Kerr
$g_{tt}$	-1	$-1 + 2 \frac{m}{r}$	$-1 + 2 \frac{mr}{\rho^2}$
$g_{rr}$	+1	$\frac{r}{r - 2m}$	$\frac{\rho^2}{\triangle}$
$g_{\theta\theta}$	$r^2$	$r^2$	$\rho^2$
$g_{\phi\phi}$	$r^2 \sin^2(\theta)$	$r^2 \sin^2(\theta)$	$(r^2 + a^2 + \frac{2 m r a^2 \sin^2(\theta)}{\rho^2}) \sin^2{\theta}$
$g_{ij}$ $i \neq j$	all zero	all zero	all zero except $g_{t \phi} = g_{\phi t} = - \frac{2 m r a \sin^2(\theta)}{\rho^2}$