We need to be precise here. Consider the following category $\mathcal{M}$:
- The objects are small categories with chosen initial object, binary coproducts, coequalisers, terminal object, binary products, and equalisers.
- The morphisms are functors that strictly preserve the chosen colimits and limits.
With a bit of work one can verify that $\mathcal{M}$ is a locally presentable category – specifically, it is clear that $\mathcal{M}$ has colimits of small filtered diagrams, limits of small diagrams, and has a small generating set. (In fact, $\mathcal{M}$ is locally finitely presentable – for this, one can check that it is the category of models for a finitary essentially algebraic theory.)
In particular, $\mathcal{M}$ has an initial object, which may be constructed using Freyd's initial object lemma.
Of course, you might object that $\mathcal{M}$ is not really the category you had in mind. Indeed, it seems more likely that you are interested in the following 2-category $\mathfrak{K}$:
- The objects are small categories with limits and colimits of finite diagrams.
- The morphisms are functors that preserve limits and colimits of finite diagrams (up to isomorphism).
- The 2-cells are natural transformations.
There is an evident forgetful functor $\mathcal{M} \to \mathfrak{K}$, and it is straightforward to check that it preserves initial objects in the following sense: if $M$ is an initial object in $\mathcal{M}$ and $K$ is an object in $\mathfrak{K}$, then the hom-category $\mathfrak{K} (U M, K)$ is equivalent to $\mathbf{1}$.
In particular, $\mathfrak{K}$ has an initial object in the sense of bicategories. However, $\mathfrak{K}$ does not have an initial object in the sense of ordinary categories.