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The composition of morphisms in a monoidal category corresponds to an associative binary operation

For a monoidal category M each morphism in M(m, m) corresponds to the operation defined for the associated monoidal set on one of the elements.

For example, in the set of natural numbers \mathbb{N} there will be a morphism corresponding to each partially-applied operation {+0, +1, ..., +n}.

Composing any two of these morphisms corresponds to performing the operation - the operation is associative because composition of morphisms is associative.