Does the bicategory of additive categories admit bicolimits?

Question

By bicolimit I mean what Kelly means in its "Elementary observations on 2-categorical limits". If we have a diagram (pseudofunctor) $G\colon\mathcal P\to\mathcal K$, the bicolimit of $G$ is the (unique up to equivalence) object $\mathsf{bicolim}G$ in $\mathcal K$ such that for every object $A$ we have a natural equivalence of categories

$$\mathcal K(\mathsf{bicolim}G,A)\simeq\mathsf{PsNat}(\Delta\ast,\mathcal K(G-,A)),$$

where $\mathsf{PsNat}$ is the category having modifications as 1-cells, and $\Delta\ast$ is the constant pseudofunctor $\mathcal P^\mathsf{op}\to\mathsf{Cat}$. Now my aim is to prove that for $\mathcal K=\mathsf{Add}$, the bicategory of additive categories, additive functors and natural transformations is indeed bicocomplete. Kelly proves this for $\mathsf{Cat}$ with an argument that leads to something more, that is an isomorphism of the above categories. This fact, and the feeling that I suppose the equivalence not to be necessary an isomorphism for $\mathsf{Add}$, suggests me that we cannot use the same argument in this context.

Are there explicit constructions of a bicolimit for categories that we can mimick in the additive case? Or, even better, are there some general results that may turn applicable to this setting?

Thank you so much!

Answer 1

Yes, additive categories are $Ab$-enriched categories with all finite direct sums. $Ab$-enriched categories have all 2-limits and 2-colimits by enriched category theory and in particular all bilimits and bicolimits. Additive categories are bireflective inside $Ab$-enriched categories, so they have bilimits and bicolimits as well.