The 1-category of small $A_\infty$ categories is certainly cocomplete, as it is the category of representations of a Lawvere algebraic theory and hence presentable. But this is, of course, not the category that you care about. Presumably, you care instead about some version of an $(\infty,2)$-category, since $A_\infty$ categories are some version of $(\infty,1)$-categories.
– Theo Johnson-FreydApr 20 '13 at 5:43

I actually do want the plain 1-category. Could you please give a reference? Sounds very interesting.
– yashaApr 20 '13 at 15:00