We generalize the operadic approach to algebraic quantum field theory [arXiv:1709.08657] to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also develop a derived version of the linear quantization adjunction for chain complex valued field theories, which in particular defines a homotopically meaningful quantization prescription for linear gauge theories.