For a Gaussian process, how does $\mathcal{N}(\mathbf{f}, {\sigma_n}^2\mathbf{I})$ relate to the Normal PDF?

Question

I am told during the Gaussian Process class that the following notations are equivalent as convention: \begin{align*} \mathbf{y}|\mathbf{f} &\sim \mathcal{N}(\mathbf{f}, {\sigma_n}^2\mathbf{I}) \\ p(\mathbf{y}|\mathbf{f}) &= \mathcal{N}(\mathbf{f}, {\sigma_n}^2\mathbf{I}) \end{align*}

How does it relate to the typical definition of pdf of a Gaussian distribution:

Normal Probability Density Function $$F(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$