Motivated by the challenge of defining twisted quantum field theories in thecontext of higher categories, we develop a general framework for lax and oplaxtransformations and their higher analogs between strong $(\infty, n)$-functors.We construct a double $(\infty,n)$-category built out of the target $(\infty,n)$-category governing the desired diagrammatics. We define (op)laxtransformations as functors into parts thereof, and an (op)lax twisted fieldtheory to be a symmetric monoidal (op)lax natural transformation between fieldtheories. We verify that lax trivially-twisted relative field theories are thesame as absolute field theories. As a second application, we extend the higherMorita category of $E_d$-algebras in a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ to an $(\infty, n+d)$-category using the highermorphisms in $\mathcal{C}$.
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