nLab lax F-natural transformation

Definition

Let $K$ and $L$ be F-categories (strict for simplicity), with 2-categories $K_\tau,L_\tau$ of tight morphisms and $K_\lambda,L_\lambda$ of loose morphisms. Let $F,G:K\to L$ be $\mathcal{F}$-functors (also strict for simplicity).

A pseudo/lax $\mathcal{F}$-natural transformation $\alpha : F \to G$ consists of:

• A lax natural transformation $\alpha_\lambda : F_\lambda \to G_\lambda$,
• The 1-morphism components of $\alpha_\lambda$ are tight, and
• The 2-morphism naturality constraint of $\alpha_\lambda$ at any tight arrow of $K$ is an isomorphism.

In particular, $\alpha_\lambda$ restricts to a pseudo natural transformation $\alpha_\tau : F_\tau \to G_\tau$.

Similarly, an $\mathcal{F}$-natural transformation can be strict/lax, pseudo/oplax, and so on.

Examples

• If $K$ and $L$ are chordate (all morphisms are tight), then a pseudo/lax transformation is simply a pseudo transformation.

• If $L$ is chordate but $K$ is inchordate (only identities are tight), then a pseudo/lax transformation is simply a lax transformation.

• If $L$ is inchordate, there are no nonidentity pseudo/lax transformations.

Applications

• Lax $\mathcal{F}$-natural transformations appear in the notion of lax F-adjunction.