In Carroll's Introduction to General Relativity: Spacetime and Geometry, there is a section titled Classical Field Theory in chapter 1. There, he mentions that:
"The action leads via a direct procedure (involving varying with respect to the metric itself) to a unique energy-momentum tensor. Applying this procedure to $$L = -\frac{1}{2}\eta^{\mu \nu}(\partial_\mu \phi)(\partial_\nu \phi) - V(\phi)$$ leads straight to the energy momentum tensor for a scalar field theory, $$T^{\mu \nu}_{scalar} = \eta^{\mu \lambda}\eta^{\nu \sigma}\partial_{\lambda}\phi \partial_{\sigma}\phi - \eta^{\mu \nu}[\frac{1}{2}\eta^{\lambda \sigma}\partial_{\lambda}\phi \partial_{\sigma}\phi + V(\phi)]."$$
How is this last expression obtained?