Consider a proper simplicial combinatorial model category $\mathcal{M}$. Consider a functor $F:I\to J$ between small categories. It induces a right Quillen functor $F^*:\mathcal{M}^J \to \mathcal{M}^I$ between the projective model structures.

Suppose that $F:I\to J$ induces a homotopy equivalence between the classifying spaces of $I$ and $J$. Can we say something more about $F^*:\mathcal{M}^J \to \mathcal{M}^I$ or its left adjoint ?

Consider a fixed proper simplicial combinatorial model category $\mathcal{M}$. Consider a functor $F:I\to J$ between small categories. It induces a right Quillen functor $F^*:\mathcal{M}^J \to \mathcal{M}^I$ between the projective model structures.

Suppose that $F:I\to J$ induces a homotopy equivalence between the classifying spaces of $I$ and $J$. Can we say something more about $F^*:\mathcal{M}^J \to \mathcal{M}^I$ or its left adjoint ?