(Op)lax natural transformations, relative field theories, and the "even higher" Morita category of $E_d$-algebras

Motivated by the challenge of defining relative, also called twisted, quantum field theories in the context of higher categories, we develop a general framework for both lax and oplax transformations and their higher analogs between strong $(\infty, n)$-functors. Namely, we construct a double $(\infty,n)$-category built out of the target $(\infty, n)$-category that we call its (op)lax square, which governs the desired diagrammatics. Both lax and oplax transformations are functors into parts thereof. We then define a lax or oplax relative field theory to be a symmetric monoidal lax or oplax natural transformation between field theories. We verify in particular that lax trivially-twisted relative field theories are the same as absolute field theories. Finally, we use the (op)lax square to extend the construction of the higher Morita category of $E_d$-algebras in a symmetric monoidal $(\infty, n)$-category $\mathcal{C}$ to an even higher level using the higher morphisms in $\mathcal{C}$.