Graduate Klein Gordon Lagrangian -- Summation question

[LagrangeEuler]

Klein Gordon Lagrangian is given by
\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^2\phi^2
I saw also this link
https://www.pas.rochester.edu/assets/pdf/undergraduate/the_free_klein_gordon_field_theory.pdf
Can someone explain me, what is
\partial_{\mu}\phi\partial^{\mu}\phi
this is some sumation so I suppose that $\mu$ is dummy index? Right? So is it correct to write
\partial_{\mu}\phi\partial^{\mu}\phi=\partial_{\alpha}\phi\partial^{\alpha}\phi?

 

[ergospherical]

That's correct, equivalently $\partial_{\mu} \phi \partial^{\mu} \phi = g^{\mu \nu} \partial_{\mu} \phi \partial_{\nu} \phi$ where as usual summation is required over repeated indices.