Implication Details

Claim: If a category has disjoint finite coproducts and is thin, then it is trivial.

Proof: For every object $A$ the two inclusions $A \rightrightarrows A + A$ must be equal, so their equalizer is $A$, but also $0$ since the coproduct is disjoint. Hence $A = 0$.

This implication has a dual.