Lemma 21.4.5. Let \(F\colon C \to D\) be a functor. Then the following conditions are equivalent:
- (1)
-
The functor \(F\) is fully faithful;
- (2)
-
For every object \(y\) of \(C\) the commutative square
is a pullback square;
- (3)
-
For every object \(x\) of \(C\) the commutative square
is a pullback square.
Proof sketch. We prove the equivalence between (1) and (2); the equivalence between (1) and (3) is dual. The two vertical functors in the commutative square in (2) are right fibrations, in a sense to be explained below in Chapter 23. One can prove that a functor between two right fibrations is an equivalence if and only if it induces equivalences on all fibers. Since the induced map on fibers is precisely the map \(\Hom _C(x,y) \to \Hom _C(Fx,Fy)\), it follows that condition (2) is equivalent to full faithfulness of \(F\). □
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