Deep networks have shown remarkable performance across a wide range of tasks, yet getting a global concept-level understanding of how they function remains a key challenge. Many post-hoc concept-based approaches have been introduced to understand their workings, yet have been shown to be not always faithful to the model. Further, they make restrictive assumptions on the concepts a model learns, such as class-specificity, small spatial extent, or alignment to human expectations. In this work, we put emphasis on the faithfulness of such concept-based explanations and propose a new model with model-inherent mechanistic concept-explanations. Our concepts are shared across classes and, from any layer, their contribution to the logit and their input-visualization can be faithfully traced. We also leverage foundation models to propose a new concept-consistency metric C2-score that can be used across different concept-based methods. We show that, compared to prior work, our concepts are quantitatively more consistent and users find our concepts to be more interpretable, all while retaining competitive ImageNet performance. Code available at https://github.com/m-parchami/FaCT.