  For a diagram of simplicial combinatorial model categories, we show that the
associated lax limit, endowed with the projective model structure, is a
presentation of the lax limit of the underlying $\infty$-categories. Our
approach can also allow for the indexing category to be simplicial, as long as
the diagram factors through its homotopy category. Analogous results for the
associated homotopy limit (and other intermediate limits) directly follow.
