Pseudo-categories (Q2381408)

If \({\mathcal C}\) is a (finitely complete) category then, for each object \(C\), the slice category \({\mathcal C}/C\times C\) becomes monoidal in such a way that the monoids are categories in \({\mathcal C}\) with object-of-object \(C\). If \({\mathcal C}\) is a (finitely complete) 2-category then the slice \({\mathcal C}/C\times C\) becomes monoidal as a 2-category; the pseudo-categories in \({\mathcal C}\) (with object-of-object \(C\)) are the pseudomonoids in this monoidal 2-category. All the higher morphisms of these structures are defined: in particular, for two pseudo-categories \(\mathbb C\) and \(\mathbb D\) in \(\mathcal C\), a pseudo-category \(\text{PsCat}(\mathbb C,\mathbb D)\) in \({\mathcal C}at\) is obtained. Pseudo-categories in \({\mathcal C}at\) generalize both bicategories and double categories.