Let us consider the ordinary category of $(2,1)$-categories. Its objects are groupoid enriched categories, and its morphisms are 2-functors. Is there an explicit way to define the objects, morphisms and 2-cells of the pullback?
So suppose we are given $G:E\to D$ and $F:C\to D$ two 2-functors, and $Z$ its pullback. It is natural to think that the objects of this $(2,1)$-category are given by the pairs $(X,Y)$ such that $F(X)=G(Y)$, and maybe that the 1-cells given by morphisms $(f,g):(X,Y)\to (X',Y')$ such that $F(f)\cong F(g)$ connected by a defined 2-cell. How should we think about the 2-cells between such pairs of morphisms? A pair of two cells $H,H'$ such that they commute in the natural way with the corresponding 2-cells between $F(f)$ and $G(f)$, and $F(f')$ and $G(g')$? Or is there another description?
Thanks!
The pullback just has as objects the set pullback of the objects of $E$ and $C$ over $D$, and similarly for the morphisms and 2-morphisms. So if $G(e)=F(d)$ and $G(e')=F(d')$ then the morphisms from $(e,d)$ to $(e',d')$ in the pullback are the morphisms $(f,f')$ such that $f:e\to e',g:d\to d',$ and $G(f)=F(f')$; the 2-morphisms $(f,f')\to (g,g')$ are the pairs $(\alpha,\alpha')$ where $\alpha:f\to f',\alpha':g\to g'$, and $G(\alpha)=F(\alpha')$.
This is not too exciting but is manageable; it's not really the right notion for most purposes, though. More often we work in a 2-category of (2,1)-categories and then we're likely interested in the pseudo-pullback, aka homotopy pullback, 2-pullback, etc. Here the objects are triples $(e,c,f:G(e)\cong F(c))$ and the morphisms and 2-morphisms must all cohere with the given isomorphisms.