adjoint functor theorem
adjoint lifting theorem
small object argument
Freyd-Mitchell embedding theorem
relation between type theory and category theory
sheaf and topos theory
enriched category theory
higher category theory
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A full and faithful functor is a functor which is both full and faithful. “Full and faithful” is sometimes shortened to “fully faithful” or “ff.” See also full subcategory.
A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits.
This is evident from inspection of the defining universal property.
There is a bigger pattern at work here which is indicated at stuff, structure, property and k-surjective functor.
For (∞,1)-categories the corresponding notion of fully faithful functor is described at
essentially surjective functor
equivalence of categories
full and faithful (∞,1)-functor