Smash product of pointed objects in lextensive categories

We describe a sufficient condition on a finitely complete and cocompletelextensive category X, under which the categorical smash product provides a canonical (symmetric, distributive with respect to finite coproducts) monoidal structure on the category (1↓X) of its pointed objects. We also show that the ground category can be reconstructed as the category of objects with counit in (1↓X).