The Clowder Project

Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors and let $I\colon \mathcal{X}\to \mathcal{C}$ be a functor.

1. Interaction With Right Whiskering. If $I$ is full and dominant, then the map
   \[ -\mathbin {\star }\text{id}_{I} \colon \text{Nat}\webleft (F,G\webright )\to \text{Nat}\webleft (F\circ I,G\circ I\webright ) \]

   is a bijection.

2. Interaction With Adjunctions. Let $\webleft (F,G\webright )\colon \mathcal{C}\mathbin {\rightleftarrows }\mathcal{D}$ be an adjunction.
   1. If $F$ is dominant, then $G$ is faithful.
   2. The following conditions are equivalent:
      1. The functor $G$ is full.
      2. The restriction
         \[ \left.G\right\vert _{\mathrm{Im}_{F}}\colon \mathrm{Im}\webleft (F\webright )\to \mathcal{C} \]

         of $G$ to $\mathrm{Im}\webleft (F\webright )$ is full.