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This is only a partial answer to your questions, but one that I imagine recovers all the cases of interest (in particular it gives a conceptual reason why your examples hold when $j$ is the Yoneda emb …

It is certainly the case that the duals of internal homs have appeared significantly less in the categorical literature. I've included a few more references below, but I am not sure this is a satisfyi …

Most Kan extensions arising in nature are pointwise, and this observation prompts Kelly to write [1]: Our present choice of nomenclature is based on our failure to find a single instance where a [non …

I'm not sure I've followed your question exactly, so let me rephrase it to how I've understood it, and you can tell me if I've misunderstood. I shall use different letters to make sure I'm not acciden …

Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that fo …

A partial answer is contained in Betti's Formal theory of internal categories (page 49), where he states that the bicategory $\mathbf{Dist}(\mathcal E)$ of $\mathcal E$-internal distributors is the fr …

A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-close …