It is well known that reflective subcategories of complete categories are complete, and that limits in the subcategory are computed by taking the limit in the ambient category and applying the reflector (however it will act trivially, see the comments). Has this been proven yet for $(\infty,1)$-categories? I know that if the ambient $(\infty,1)$-category is (locally) presentable, and the subcategory is accessible that this is in HTT, however this is a very special case, and the latter condition is often hard to verify even when dealing with the presentable case. Has anything been worked out on this?