Fully faithful functor

Let $F:\mathcal C\to\mathcal D$ be a functor. It is fully faithful if, for every pair of objects $X,Y$ in $\mathcal C$ , the induced map

\operatorname{Hom}_{\mathcal C}(X,Y)\longrightarrow \operatorname{Hom}_{\mathcal D}(F(X),F(Y)),\qquad f\longmapsto F(f)

is a bijection.

Consequences

A fully faithful functor is faithful and is full: every morphism $F(X)\to F(Y)$ is $F(f)$ for a unique morphism $f:X\to Y$ .

Example

The inclusion of a full subcategory into its ambient category is fully faithful.