# A complete category of categories and embeddings

I have a theory that you should be able to construct a category of categories and embeddings that is complete. Someone already pointed out that if the functors are full and faithful, your cat is not complete. Can we choose a type of embedding that ensures that the cat is complete? My intuition is coming from "just arrow categories" as seen here. In that post, I am trying to find the definiton of a functor and the responder has suggested the definition of subcategory for "just arrow" cats. Am I wrong in thinking that there is an embedding (not full faithful) that constructs a complete cat? What if we reverse the arrows and consider quotients in the "just arrow" categories?

As you mentioned in your previous questions, $\mathbf{Cat}$ is equivalent to $JA-\mathbf{Cat}$. To understand anything categorical about $JA-\mathbf{Cat}$ is to understand it about $\mathbf{Cat}$, because all categorical properties are stable under equivalence.
However, the wide subcategory of $\mathbf{Cat}$, whose arrows are (any types of) embeddings is not complete. The obvious reason is that it has no terminal object. So is the category $JA-\mathbf{Cat}$.
Regarding your question about $JA$-quotients: yes, $JA-\mathbf{Cat}$ has quotients, but they are much more sophisticated, then subcategories, as well as in $\mathbf{Cat}$. See Generalized congruences -- Epimorphisms in Cat for description of epimorphisms.