**Puzzle 226** Here's some answers

+ The category of groups, group homomorphisms, with the direct sum (coproduct) as the monoidal product and the trivial group as the unit

+ Monoidal posets: a monoidal category where every homset is either a singleton or empty.

+ The category of set with the disjoint union as the monoidal product and singletons as the unit.

But I had this conjecture while thinking of examples. Every example was basically using a coproduct as the monoidal product.

**Conjecture**: Any category can be turned into a monoidal category if it has all coproducts and a unit object \\(I\\) such that \\(\hom(I,I) = \lbrace1_I\rbrace\\) (it only has one morphism from itself to itself, the identity map), where the monoidal product is given by the coproduct and the unit object is the monoidal unit.

I'll try and prove this later.

+ The category of groups, group homomorphisms, with the direct sum (coproduct) as the monoidal product and the trivial group as the unit

+ Monoidal posets: a monoidal category where every homset is either a singleton or empty.

+ The category of set with the disjoint union as the monoidal product and singletons as the unit.

But I had this conjecture while thinking of examples. Every example was basically using a coproduct as the monoidal product.

**Conjecture**: Any category can be turned into a monoidal category if it has all coproducts and a unit object \\(I\\) such that \\(\hom(I,I) = \lbrace1_I\rbrace\\) (it only has one morphism from itself to itself, the identity map), where the monoidal product is given by the coproduct and the unit object is the monoidal unit.

I'll try and prove this later.