||Multi-strain competition on networks is observed in many contexts, including infectious disease ecology, information dissemination or behavioral adaptation to epidemics. Despite a substantial body of research has been developed considering static, time-aggregated networks, it remains a challenge to understand the transmission of concurrent strains when links of the network are created and destroyed over time. Here we analyze how network dynamics shapes the outcome of the competition between an initially endemic strain and an emerging one, when both strains follow a susceptible-infected-susceptible dynamics, and spread at time scales comparable with the network evolution one. Using time-resolved data of close-proximity interactions between patients admitted to a hospital and medical health care workers, we analyze the impact of temporal patterns and initial conditions on the dominance diagram and coexistence time. We find that strong variations in activity volume cause the probability that the emerging strain replaces the endemic one to be highly sensitive to the time of emergence. The temporal structure of the network shapes the dominance diagram, with significant variations in the replacement probability (for a given set of epidemiological parameters) observed from the empirical network and a randomized version of it. Our work contributes towards the description of the complex interplay between competing pathogens on temporal networks.