Recall that a poset $K$ is a complete lattice if and only if $K$ is injective with respect to poset embeddings in that sense that for any poset $B$, any embedded subposet $A \subseteq B$, and any order-preserving map $A \to K$, there exists an extension $B \to K$.
I'm wondering what sorts of generalizations or analogs this fact has when passing from posets to categories. Ultimately I'd be interested to know if (co)completeness conditions can be characterized by injectivity, but I'm not sure what the correct question is in this direction, so let's start with something concrete:
Question: Which locally small categories $\mathcal K$ are injective with respect to fully faithful functors $\mathcal A \to \mathcal B$ between small categories?
There is some ambiguity about what "injective" means here. In general, "injective with respect to fully faithful functors" means that given a fully faithful functor $\mathcal A \to \mathcal B$ and a functor $\mathcal A \to \mathcal K$, there exists a functor $\mathcal B \to \mathcal K$ making the requisite diagram commute. But I can think of at least 4 possible meanings of "commute" -- we could ask for the diagram to commute strictly (giving (1) "strict-injectivity"), up to isomorphism (giving (2) "pseudo-injectivity"), or up to a natural transformation (giving (3) "lax-injectivity" or (4) "oplax-injectivity" depending on the direction of the transformation). So really there are at least 4 questions here. I think the most interesting versions are the "strict" and "pseudo" versions, and I suspect the answers in these two cases should be rather close.
At any rate, it seems that by duality, the answer can't be "the complete categories" except perhaps in case (3) or (4).
If it makes a difference to change from considering fully faithful functors $\mathcal A \to \mathcal B$ to just considering full replete subcategories $\mathcal A \subseteq \mathcal B$ or something like that, I'd be happy with an answer to any such small variation.
I would also be interested in knowing the answer when considering fully faithful functors between locally small categories, rather than just between small categories.