Implication Details

Claim: If a category has an initial object and is left cancellative, then it has a strict initial object.

Proof: It suffices to prove that in general any monomorphism $f : A \to 0$ into an initial object is an isomorphism. If $g : 0 \to A$ is the unique morphism, then $f \circ g = \id_0$ since $0$ is initial. But then $f$ is a split epimorphism and a monomorphism, hence an isomorphism.

This implication has a dual.