Given a displayed category over $\mathrm{𝒞︀}$ and an object $𝑐:{\mathrm{𝒞︀}}_0$, the fiber of over $𝑐$ is the category whose objects are the displayed objects , and whose morphisms are the vertical morphisms , those lying over the identity.

Identities are the displayed identities. The displayed composite of two vertical morphisms lies over ${\text{𝗂𝖽}}_{𝑐}\star {\text{𝗂𝖽}}_{𝑐}$ rather than ${\text{𝗂𝖽}}_{𝑐}$, so composition in the fiber transports it along ${\text{𝗂𝖽𝖫}}_{{\text{𝗂𝖽}}_{𝑐}}$.