This work is dedicated to the study of composition operator Cφ where φ is a continuous function. First, we characterize topologies on Cθ(X,Y), the space of θ-faintly continuous, to establish results for the composition operator Cφ, acting between spaces of θ-faintly continuous functions. Next, topologies on Cα(X,Y), the space of α-faintly continuous, are analyzed in the same way to obtain results again for the composition operator Cφ. In both cases, weak, strong and exponential topologies are investigated, obtaining inclusion properties and where the evaluation map is key.