  We delve into the concept of categories with products that distribute over
coproducts, which we call doubly-infinitary distributive categories. We show
various instances of doubly-infinitary distributive categories aiming for a
comparative analysis with established notions such as extensivity, infinitary
distributiveness, and cartesian closedness. Our exploration reveals that this
condition represents a substantial extension beyond the classical understanding
of infinitary distributive categories. Our main theorem establishes that free
doubly-infinitary distributive categories are cartesian closed. We end the
paper with remarks on non-canonical isomorphisms, open questions, and future
work.
