  Building on the foundations in our previous paper, we study Segal conditions
that are given by finite products, determined by structures we call cartesian
patterns. We set up Day convolution on presheaves in this setting and use it to
give conditions under which there is a colimit formula for free algebras and
other left adjoints. This specializes to give a simple proof of Lurie's results
on operadic left Kan extensions and free algebras for symmetric
$\infty$-operads.
