(It is asked in http://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.)

Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a pseudofunctor (other type of 2-functors may be considered as well) between bicategories. Suppose that $\mathcal{C}$ has all limits.

For fixed $A\in \mathcal{A}$, we compute the pseudolimit of $F(A,-)$ respect to $\mathcal{B}$. The limit is not unique but well-defined up to equivalences and the equivalences are unique up to a unique 2-cell.

The question:

1. can we produce a pseudofunctor $\mathcal{A}\to \mathcal{C}$, uniquely defined up to natural transformations, and the natural transformations are unique up to a unique modification?

2. For the induced pseudofunctor, we compute the limit again. Are the limits commutative?

This may be standard facts; for 1-category one can verify it by hand (see also Mac Lane).