The multiplication πœ‡:π‘€βˆ˜π‘€β†’π‘€ of a monad 𝑀:π’žβ†’π’ž allows one to compose 𝑓:𝐴→𝑀𝐡 and 𝑔:𝐡→𝑀𝐢 via

π‘“π‘€π‘”πœ‡πΆπ΄π‘€π΅π‘€(𝑀𝐢)𝑀𝐢

However, if includes context in, then the target to compose are

  1. 𝑓:Γ×𝐴→𝑀𝐡
  2. 𝑔:Γ×𝐡→𝑀𝐢

we need a natural transformation

𝑠Γ,𝐡:Γ×𝑀𝐡→𝑀(Γ×𝐡)

with which the composite

(πœ‹1,𝑓)𝑠Γ,π΅π‘€π‘”πœ‡πΆΞ“Γ—π΄Ξ“Γ—π‘€π΅π‘€(Γ×𝐡)𝑀(𝑀𝐢)𝑀𝐢

To make the composition associative and pure computation (πœ‚π΄Β·πœ‹2):Γ×𝐴→𝑀𝐴 an identity, 𝑠 must satisfy certain coherence condition 1.