The Stacks project

Remark 85.4.1. In Situation 85.3.3 an augmentation $a_0$ towards a site $\mathcal{D}$ will mean

1. $a_0 : \mathcal{C}_0 \to \mathcal{D}$ is a morphism of sites given by a continuous functor $u_0 : \mathcal{D} \to \mathcal{C}_0$ such that for all $\varphi , \psi : [0] \to [n]$ we have $u_\varphi \circ u_0 = u_\psi \circ u_0$.

2. $a_0 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_0) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ is a morphism of topoi given by a cocontinuous functor $u_0 : \mathcal{C}_0 \to \mathcal{D}$ such that for all $\varphi , \psi : [0] \to [n]$ we have $u_0 \circ u_\varphi = u_0 \circ u_\psi$.