On faithful conditional identities and conditionally complete conditional varieties

It is proved that for any faithful conditional variety $\mathfrak M$ with unique constant algebra there exist a conditionally complete conditional variety $\mathfrak M^{\mathrm c}$ and a polyinjective functor $F$ which isomorphically embedds the embedding category $\overset{\rightarrowtail}{\mathfrak M}$ into the embedding category $\overset{\rightarrowtail}{\mathfrak M^{\mathrm c}}$.