In 1-category theory a representation of a 1-functor $F:C\to Set$ is a 0-cell $X$ in $C$ together with a universal element $u\in FX$ such that the transformation $C(-,X)\to F$ is an isomorphism (=a 1-equality) in the 1-category of functors $Fun(C,Set)$. It is well known how unique a representation $(X,u)$ is: unique up to a unique compatible isomorphism.

To see the uniqueness of the representation better, one can unfold $F$ into its category of elements $el(F)$. The representations of $F$ are then precisely the initial objects of $el(F)$. When we consider only the full subcategory $rep(F)$ of representations of $F$ in $el(F)$, and if we consider the unique functor $rep(F)\to \mathbf 1$, the one point category, then we see that this functor is an equivalence. This means that $rep(F)$ is contractible and the representation is unique up to contractibility.

Can we mimic this in the (weak) 2-categorical setting?

All my $2$-categories, functors and transformations are per default weak (but not lax). A representation of a $2$-functor $F:C\to Cat$ is a pair $(X,u)$, where $X$ is a $0$-cell of $C$ and $u$ an object of $F X$, such that the transformation of 2-functors $$C(X,-)\to F$$ induced by the 2-Yoneda lemma is an equivalence in the $2$-category $2Cat(C,Cat)$. How unique is a representation of $F$?

Is it possible to unfold $F$ into a (weak) 2-category $el(F)$ such hat the representations are precisely (weak) 2-initial objects in that 2-category, and such that the full sub-2-category $rep(F)$ of representations is contractible? By this I mean that the 2-functor $rep(F)\to \mathbf 1$ into the one-point 2-category is a 2-equivalence.

In 1-category theory a representation of a 1-functor $F:C\to Set$ is a 0-cell $X$ in $C$ together with a universal element $u\in FX$ such that the transformation $C(X,-)\to F$ is an isomorphism (=a 1-equality) in the 1-category of functors $Fun(C,Set)$. It is well known how unique a representation $(X,u)$ is: unique up to a unique compatible isomorphism.

To see the uniqueness of the representation better, one can unfold $F$ into its category of elements $el(F)$. The representations of $F$ are then precisely the initial objects of $el(F)$. When we consider only the full subcategory $rep(F)$ of representations of $F$ in $el(F)$, and if we consider the unique functor $rep(F)\to \mathbf 1$, the one point category, then we see that this functor is an equivalence. This means that $rep(F)$ is contractible and the representation is unique up to contractibility.

Can we mimic this in the (weak) 2-categorical setting?

All my $2$-categories, functors and transformations are per default weak (but not lax). A representation of a $2$-functor $F:C\to Cat$ is a pair $(X,u)$, where $X$ is a $0$-cell of $C$ and $u$ an object of $F X$, such that the transformation of 2-functors $$C(X,-)\to F$$ induced by the 2-Yoneda lemma is an equivalence in the $2$-category $2Cat(C,Cat)$. How unique is a representation of $F$?

Is it possible to unfold $F$ into a (weak) 2-category $el(F)$ such hat the representations are precisely (weak) 2-initial objects in that 2-category, and such that the full sub-2-category $rep(F)$ of representations is contractible? By this I mean that the 2-functor $rep(F)\to \mathbf 1$ into the one-point 2-category is a 2-equivalence.