By summing the totient of every number up to 100 we get the number of ordered coprime pairs (including $(1,1)$)

$\displaystyle \sum_{n=1}^{100} \phi(n) = 3044$

Now, to count the reverse of each ordered pair, we multiply by $2$. We also have to subtract one from this since $(1,1)$ is counted for twice otherwise.

$\displaystyle 2 \left(\sum_{n=1}^{100} \phi(n)\right) -1 = 6087$

This Desmos graph is nice to work with, but I don't know who made it ( CoprimePairStuff )