  In this paper we prove that various quasi-categories whose objects are
$\infty$-categories in a very general sense are complete: admitting limits
indexed by all simplicial sets. This result and others of a similar flavor
follow from a general theorem in which we characterize the data that is
required to define a limit cone in a quasi-category constructed as a homotopy
coherent nerve. Since all quasi-categories arise this way up to equivalence,
this analysis covers the general case. Namely, we show that quasi-categorical
limit cones may be modeled at the point-set level by pseudo homotopy limit
cones, whose shape is governed by the weight for pseudo limits over a homotopy
coherent diagram but with the defining universal property up to equivalence,
rather than isomorphism, of mapping spaces. Our applications follow from the
fact that the $(\infty,1)$-categorical core of an $\infty$-cosmos admits
weighted homotopy limits for all flexible weights, which includes in particular
the weight for pseudo cones.
