Edit: rewritten the question (edit:again), as I realised I wanted weak pseudo-pullbacks, not comma objects.


Consider the 2-category $V$-$Cat$ of $V$-enriched categories. An example is $Cat$ itself, where we take $V=Cat$. I'm interested in the more general setting where $V$ is at least finitely complete and complete, if not possessing small limits and colimits outright. I also know that $V$ is such that small $V$-categories can be interpreted as categories internal to $V$ with object of objects given by a coproduct of the tensor unit.


Pseudo-pullbacks exist in $Cat$, and can be calculated as a strict finite limit $A\times\_C C^I \times\_C B$ where $C^I$ is the isomorphism category of $C$. Actually I may not be interested in the pseudo-pullbacks, but strict 2-pullbacks, depending on what this example in $Cat$ is.


Based on Finn's answer, I guess this may exist when cotensors by $I$ exist ($I$ is the groupoid with two objects $a,b$ with a unique isomorphism $a \stackrel{\sim}{\to} b$). So this then is my question:



> 
> When do pseudo/strict 2-pullbacks (as appropriate) exist in $V$-$Cat$?
> 
> 
> 



When does the 2-category V-Cat have pseudo-pullbacks? 
As I mentioned in the comment above, "weak limit" is normally defined as for limit, but with the universal property modified to ask only for existence not uniqueness. The 2-dimensional limit notion in which all equations between 1-cells are replaced by suitably coherent invertible 2-cells is usually given the prefix "bi". 


Any 2-category with finite limits (in the strict 2-categorical sense) also has isocomma objects (defined like comma objects but with an invertible 2-cell), pseudopullbacks, and bipullbacks. 


If V has finite limits (in the ordinary sense) then V-Cat has finite limits (in the strict
2-categorical sense). 


(The parts of this that relate to the original version of your question are dealt with in Finn's answer.)


