$p$-adic Langlands functoriality for the definite unitary group

We formalise a notion of $p$-adic Langlands functoriality for the definite unitary group. This extends the classical notion of Langlands functoriality to the setting of eigenvarieties. We apply some results of Chenevier to obtain some cases of $p$-adic Langlands functoriality by interpolating known cases of classical Langlands functoriality.