Item 2 (011D) — The Clowder Project

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*To be replaced with Linus Romer's Elemaints when it is released.

Interaction With Precomposition I. Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
1. If $F$ is faithful, then the precomposition functor
  \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  can fail to be faithful.
2. Conversely, if the precomposition functor
  \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  is faithful, then $F$ can fail to be faithful.

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