On the nLab page for sieves and elsewhere it is asserted that a fully faithful functor is 'equivalent' to the inclusion functor of a full subcategory -- what is this intended to mean, explicitly?
Are there equivalences in the $2$-category of functors, natural transformations and modifications between fully faithful functors and inclusions of full subcategories? Is there an equivalence of categories between the domain category of a fully faithful functor and a full subcategory of the codomain category? Concretely:
What does it mean non-heuristically that a fully faithful functor can be thought of as a full subcategory?
I am trying to understand sieves and all the literature I've found defines them as fully faithful discrete fibrations then immediately begins discussing them as full subcategories closed under precomposition, and I'm missing the exact connection between these two interpretations.
It's clear that for any functor $F:\mathcal{C}\to\mathcal{D}$ we have $F(\mathcal{C})$ as a subcategory of $\mathcal{D}$ and $F(\mathcal{C})$ is a full subcategory iff $F$ is full; is this related to the intended meaning? This is incorrect, thanks to Jendrik for catching the error in the comments.