Let $\mathcal I$, $\mathcal C$ be $2$-categories (or $(\infty, 2)$-categories, I'm interested in both cases) and assume that $\mathcal I$ is small, $\mathcal C$ has enough weighted (co)limits as you need. We can define the $2$-category of (co)lax functors $\operatorname{Fun}^\text{(co)lax}(\mathcal I, \mathcal C)$. (**Edit**: I wrote an incorrect description for this which is now deleted)


Is there a nice way to describe (co)limits in (co)lax functor categories? I don't expect it's too simple like the non-lax case (which is computed pointwise) because lax functors specialize to monads, and colimits of algebras are usually complicated. But more specifically:


1. Is there a known formula for them? By this, I mean an expression for enough data (value on cells under the diagram) in terms of, say, some weighted (co)limits in $\mathcal C$.
2. Are they simple to compute in some cases, e.g., filtered colimits?
3. In particular, if $\mathcal C$ is (co)complete, can we say that $\operatorname{Fun}^\text{(co)lax}(\mathcal I, \mathcal C)$ is (co)complete?



(Co)limits in lax functor categories
It depends on what kind of morphisms between your lax functors you're interested in. (As Tim says, lax functors are not the objects of the Gray internal hom.)


For any 2-category $I$, there is a 2-category $I'$ such that lax functors out of $I$ are the same as strict functors out of $I'$. Thus, the 2-category of lax functors and *strict* natural transformations is equivalent to an ordinary $\rm Cat$-enriched functor category, and thus inherits all limits and colimits from its codomain 2-category $C$ computed pointwise. In addition, it is strictly 2-monadic and 2-comonadic over a set-indexed power of $C$.


The 2-category of lax functors and *pseudo* natural transformations between them is equivalent to the category of algebras and pseudomorphisms for this 2-monad or 2-comonad. Therefore, by the classical work of Blackwell-Kelly-Power on 2-monads (*Two-dimensional monad theory*), it has PIE-limits (using the monad) and PIE-colimits (using the comonad), again computed pointwise, and in particular all pseudolimits and pseudocolimits.


Similarly, the 2-categories of lax functors and lax or colax transformations are the categories of algebras and lax or colax morphisms. Such categories don't in general have even all pseudolimits or pseudocolimits, but there are significant classes of limits and colimits that they do have -- again computed pointwise. Steve Lack initiated this study in his paper [Limits for lax morphisms](http://web.science.mq.edu.au/%7Eslack/papers/talgl.pdf), and later he and I characterized precisely the class of limits that such 2-categories admit in our paper [Enhanced 2-categories and limits for lax morphisms](https://arxiv.org/abs/1104.2111).


Note that all of these limits and colimits *are* computed pointwise. It's true that colimits of monads *on a fixed category* are complicated, but here we are talking about (co)limits of monads *in a fixed 2-category* which are a different beast, and in particular involve changing the object *on* which the monad lives.


