Fully Faithful Functor

Let $\mathcal{C},\mathcal{D}$ be categories. A functor $F:\mathcal{C}\Rightarrow \mathcal{D}$ is fully faithful iff $F$ is Bijection on homsets:

$$F_{x,y}:\mathcal{C}[x,y]\cong \mathcal{D}[Fx,Fy]$$

For every pair of objects $x,y\in \mathcal{C}$.

See also

Faithful Functor Full Functor