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.
If $F:C\to D$ is a fully faithful functor and $D_0$ is the full subcategory of $D$ on the objects in the image of $F$, then $F$ restricts to an equivalence of categories $C\to D_0$. Conversely, the inclusion functor $D_0\to D$ is fully faithful, and thus so is its composition with any equivalence of categories $C\to D_0$. So a functor is fully faithful iff it is the composition of an equivalence of categories followed by the inclusion of a full subcategory. For most purposes (more precisely, any purposes that are unaffected by precomposing with an equivalence of categories), this means you can assume any fully faithful functor actually is just the inclusion of a full subcategory.