A salient merit of processing quantum information is the ability of simultaneously working with both bit zero and bit one. The basic unit of quantum information, the so-called qubit, is a superposition of two orthogonal(or near-orthogonal) quantum states which can be realised on distinct physical platforms. At present, no unique qubit encoding exists that is superior to all the other ones. Different labs are implementing their most convenient technique for encoding the qubit and so the network of labs becomes heterogeneous. In this paper, we consider two types of qubit encodings, one is the single-rail qubit in terms of discrete-variable (DV) states |0$\rangle$ and |1$\rangle$ which arerespectively the vacuum and the single-photon state and the other is the coherent-state qubit in terms of continuous variable (CV) states |α$\rangle$ and |−α$\rangle$ which are coherent states with equal amplitudes but opposite phases. We devise linear-optics schemes to teleport one type of qubit to the other type. More than that, our teleportation schemes are designed so that two kinds of controllers, one is capable of manipulating single-rail qubits (DV controller) whilethe other coherent-state ones (CV controller), are able to simultaneously supervise the tasks in both directions. We first propose a quantum circuit to prepare a relevant four-party pure entangled state serving as a quantum channelbetween the four participants: two teleporters and two controllers.We then detail the hybrid controlled teleportation protocols taking into account the dissipation effect caused by the presence of losses in the environment surrounding the participants.