Opposites of Kleisli categories.

If C is a category with coproducts, and T a monad on C, then it is easy to see that the Kleisli category Kl(T) will inherit the coproducts from C. On the other hand, Kleisli categories generally don't have products. Try your favorite monads on Set, e.g. lists, probability distributions. The crucial idea is that products of free algebras aren't generally free. Now dualize this construction to answer your question.

Why should you care about opposites of Kleisli categories? This seems rather contrived after all. Here are some ideas:

• Lawvere theories are essentially opposites of Kleisli categories (restricted to finite sets). This fits the context of this question well: a Lawvere theory is freely generated under products from a single sort $1$. We should generally not expect this to have coproducts.

• Opposites of Kleisli categories of continuation monads are prime examples of Selinger's control categories