Are cytotoxic T-cells in innate immunity?

No. Cytotoxic T-cells are involved in cell-mediated immunity, a type of adaptive immunity. 

The immune system is divided into innate and adaptive components, each of which has a distinct role and function. Innate immunity is present in the body at all times. Natural killer cells that are a part of innate immunity don’t need activation to attack harmful cells. On the other hand, adaptive immunity is triggered on encountering a foreign pathogen. Cytotoxic T cells, which are part of the adaptive immune response, only attack viruses when activated by an antigen. They play a pivotal role in immune defense against intracellular pathogens such as bacteria and viruses and against tumors.