Let \(\Ee\) be a category with finite limits and let \(K\) be a class of morphisms in \(\Ee\) which contains all identities and is closed under finite limits. Then \(K\) satisfies the left cancellation property. Dually, a class containing all identities and closed under finite colimits satisfies the right cancellation property.
Proof
It suffices to prove the first claim, as the second one is dual. Given morphisms \(f\colon X \to Y\) and \(g\colon Y \to Z\), the following commutative square in \(\Ar(T)\) exhibits \(f\) as a limit in \(\Ar(T)\) of the morphisms \(\id_Y\), \(gf\) and \(g\): Since \(K\) contains identities and is closed under finite limits, if \(g, gf \in K\), then \(f \in K\).
