5Step 1. Observe that ΣX is the colimit of any diagram of the form 0←X→X←X→…←X→0, and
ΣX ⊔ ΣX ⊔⊔ ΣX is the colimit of any diagram of the form 0←X→0←X→…←X→0.

Step 2. Observe that by sending some X’s to zero in such a diagram, we obtain a map
ΣX → ΣX ⊔ ΣX ⊔⊔ ΣX.

Step 3. Observe that this induces maps hom _{hC}(ΣX,Y ) ×× hom _{hC}(ΣX,Y ) → hom _{hC}(ΣX,Y ). Step
4. Observe that any zero morphism ΣX → 0 → Y induces a unit element for this operation.

Step 5. Observe that hom _{hC}(ΣΣX,Y ) admits two multiplications which are compatible. Therefore, by
the Eckman-Hilton Lemma they are the same. Consequently, this multiplication is commutative.