We describe the Muography Program to study the Colombian volcanoes. Mainly, we discuss the criteria adopted for designing, building, and commissioning MuTe: a hybrid Muon Telescope. MuTe implements a composite detection technique combining a hodoscope for particle tracking and a water Cherenkov detector for enhancing the muon-to-background-signal separation of extended air showers. Next, we discuss the detailed MuTe digital twin employed to estimate the instrument's response to the muon flux and its impact on its design and performance. The impinging muon flux calculation and its corresponding signals in the detector consider four factors with different spatial and time scales: the geomagnetic effects, the development of extensive air showers in the atmosphere, the propagation through the scanned structure and the detector response at a particular geographic point. Next, we examine the structural --mechanical and thermal-- behaviour of MuTe, its first calibration measurements, and identify the possible volcano candidate with the best observation points. In addition, MuTe incorporates particle-identification techniques to reduce the background noise sources and discrimination of fake events by a picosecond Time-of-Flight system and implement an offline machine learning framework. We found that the frontal flux adds up to 78% of the recorded events. From this total, 36% corresponds to electromagnetic components, 33% single-muon events, and 30% multiple particle detection. Finally, we discuss an optimization algorithm to improve our instrument's estimation of the volcano's internal density distribution based on the Simulated Annealing method.