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. 1.

   Interaction With Right Whiskering. If $I$ is full and dominant, then the map

   \[ -\mathbin {\star }\operatorname {\mathrm{id}}_{I} \colon \operatorname {\mathrm{Nat}}\webleft (F,G\webright )\to \operatorname {\mathrm{Nat}}\webleft (F\circ I,G\circ I\webright ) \]

   is a bijection.

2. 2.

   Interaction With Adjunctions. Let $\webleft (F,G\webright )\colon \mathcal{C}\mathbin {\rightleftarrows }\mathcal{D}$ be an adjunction.

   1. (a)

      If $F$ is dominant, then $G$ is faithful.

   2. (b)

      The following conditions are equivalent:

      1. (i)

         The functor $G$ is full.

      2. (ii)

         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.