A full and faithful $(\infty,1)$-functor $F : C \to D$ exhibits $C$ as a full sub-(∞,1)-category of $D$ and one tends to write

$F : C \hookrightarrow D$

to indicate this.

Properties

Every full and faithful $(\infty,1)$-functor is a monomorphism in (∞,1)Cat, but being a full and faithful $(\infty,1)$-functor is a stronger condition. An $(\infty,1)$-functor $F$ is a monomorphism if and only if it induces a monomorphism on hom-spaces and every equivalence $F X \simeq F Y$ is in the effective image of $F$ (see this MathOverflow question).