ScalingStacks

Definition 5.3. Define $\corr_{n}^{0}\mathrel{\mathop{:}}=\gaunt_{n}$, and for $k>0$, define $\corr_{n}^{k}$ recursively as follows. The objects are triples $(X_{0},X_{1},F)$ consisting of two gaunt $n$-categories $X_{0}$ and $X_{1}$ and a functor

$$F\colon X_{0}^{\mathrm{op}}\times X_{1}\to\corr_{n-1}^{k-1}.$$

(Here $\mathrm{op}=\rho(1,0,\dots,0)$ is the opposite obtained by reversing just the 1-morphisms of $X_{0}$.) A morphism $(X_{0},X_{1},F)\to(Y_{0},Y_{1},G)$ is a triple $(f_{0},f_{1},\alpha)$ consisting of functors $f_{0}\colon X_{0}\to Y_{0}$ and $f_{1}\colon X_{1}\to Y_{1}$ and a natural transformation

$$\alpha\colon F\to G\circ(f_{0}^{\mathrm{op}}\times f_{1}).$$