nLab lax extranatural transformation

Idea

A lax extranatural transformation is to a lax transformation as an extranatural transformation is to a natural transformation.

Definition

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ be bicategories and $P\colon\mathcal{A}\times\mathcal{B}^\mathsf{op}\times\mathcal{B}\to\mathcal{D}$ and $Q\colon\mathcal{A}\times\mathcal{C}^\mathsf{op}\times\mathcal{C}\to\mathcal{D}$ be pseudofunctors (we are going to use the invertibility of their lax functoriality constraints below). The following is [Corner 2017, Definition 2.2], but laxified:

Definition

A lax extranatural transformation $P\overset{\bullet}{\Rightarrow}Q$ from $P$ to $Q$ consists of

1. Components. For each $A\in\mathrm{Obj}(\mathcal{B}$ and each $C\in\mathrm{Obj}(C)$, a lax transformation

   $$\alpha_{(-),B,C}\colon P^B_{(-),B}\Rightarrow Q^{C}_{(-),C},$$

   called the component of $\alpha$ at $(B,C)$.

2. Lax Extranaturality Constraints I. For each $1$-morphism $g\colon B\longrightarrow B'$ of $\mathcal{B}$, a $2$-morphism

   $$\alpha_{g}\colon\alpha_{A,B',C}\circ P^{\mathsf{id}_{B'}}_{\mathsf{id}_{A},g}\Rightarrow\alpha_{A,B,C}\circ P^{g}_{\mathsf{id}_{A},\mathsf{id}_{B}}$$

   of $\mathcal{D}$ as in the diagram

   

   called the lax extranaturality constraint of $\alpha$ at $g$.

3. Lax Extranaturality Constraints II. For each $1$-morphism $h\colon C\longrightarrow C'$ of $\mathcal{D}$, a $2$-morphism

   $$\alpha_{h}\colon Q^{\biid_{C}}_{\biid_{A},h}\circ\alpha_{A,B,C}\Rightarrow Q^{h}_{\biid_{A},\biid_{C'}}\circ\alpha_{A,B,C'}$$

   of $\mathcal{D}$ as in the diagram

   

   called the lax extranaturality constraint of $\alpha$ at $h$.

【…】

References

• Alexander Corner, A Universal Characterisation of Codescent Objects (arXiv).