The Stacks project

Remark 14.3.3. By abuse of notation we sometimes write $d_ i : U_ n \to U_{n - 1}$ instead of $d^ n_ i$, and similarly for $s_ i : U_ n \to U_{n + 1}$. The relations among the morphisms $d^ n_ i$ and $s^ n_ i$ may be expressed as follows:

1. If $i < j$, then $d_ i \circ d_ j = d_{j - 1} \circ d_ i$.

2. If $i < j$, then $d_ i \circ s_ j = s_{j - 1} \circ d_ i$.

3. We have $\text{id} = d_ j \circ s_ j = d_{j + 1} \circ s_ j$.

4. If $i > j + 1$, then $d_ i \circ s_ j = s_ j \circ d_{i - 1}$.

5. If $i \leq j$, then $s_ i \circ s_ j = s_{j + 1} \circ s_ i$.

This means that whenever the compositions on both the left and the right are defined then the corresponding equality should hold.