On pages 17 and 18 of the following document: https://www.math.tifr.res.in/~sujatha/ihes.pdf, we find the following paragraph:

Let $ \mathbb{Q} \mathrm{HS}$ be the category of pure Hodge structures over $\mathbb{Q}$. There is a functor: $$ \mathcal{R}_{ \mathrm{Hodge} } \colon \mathop{\mathrm{Mot}}^{ \bullet }_{ \mathrm{num} } ( k, \mathbb{Q} ) \to \mathbb{Q} \mathrm{HS} $$ and the Hodge conjecture is equivalent to the assertion that $\mathcal{R}_{ \mathrm{Hodge} }$ is fully faithful.

My questions are:

How is the Hodge realization functor $ \mathcal{R}_{ \mathrm{Hodge} }$ explicitly defined? And how to prove explicitly that the Hodge conjecture is equivalent to the assertion that $\mathcal{R}_{\mathrm{Hodge}}$ is fully faithful?

Thanks in advance for your help.