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Remark 14.5.3. By abuse of notation we sometimes write $\delta _ i : U_{n - 1} \to U_ n$ instead of $\delta ^ n_ i$, and similarly for $\sigma _ i : U_{n + 1} \to U_ n$. The relations among the morphisms $\delta ^ n_ i$ and $\sigma ^ n_ i$ may be expressed as follows:

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

  2. If $i < j$, then $\sigma _ j \circ \delta _ i = \delta _ i \circ \sigma _{j - 1}$.

  3. We have $\text{id} = \sigma _ j \circ \delta _ j = \sigma _ j \circ \delta _{j + 1}$.

  4. If $i > j + 1$, then $\sigma _ j \circ \delta _ i = \delta _{i - 1} \circ \sigma _ j$.

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

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

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