Let ${\bf C}$ be a category, $\alpha$ an ordinal number, and $\zeta$ an associative function $\alpha \times \alpha \to \alpha$  (see note 1). Let us define a $5$-tuple $(D_o,D_h,u,v,d)$ as follows: $D_o$ is just the object class of $\bf C$, while $D_h$ is the quotient of $\coprod_{i \in \alpha} \hom({\bf C})$ by the equivalence $\mathcal R$ that amalgamates two pairs $(f,i)$ and $(g,j)$ iff $f=g={\rm id}_{\bf C}(A)$ for some $A \in {\rm Ob}({\bf C})$; $u$ and $v$ are the functions $D_h \to D_o$ taking the equivalence class $(f,i) \bmod \mathcal R$ to $s(f)$ and $t(f)$, respectively; and $d$ is the mapping $D_h \times_{v,u} D_h \to D_h$ sending a pair $((f,i)\bmod \mathcal R,(g,j) \bmod \mathcal R)$ to $(f,i) \bmod \mathcal R$, respectively to $(g,j) \bmod \mathcal R$, if $g$, respectively $f$, is an identity of $\bf C$, and to $(fg,\zeta(i,j)) \bmod \mathcal R$ otherwise.

It is clear that $(D_o,D_h,u,v,d)$ is a category, let me denote it by $\zeta \ast \bf C$. Since I don't know if  $\zeta \ast \bf C$ has already a name, I'm calling it the augmentation of $\bf C$ by $\zeta$: While looking forward to finding something more "natural", I'm using it as a toy example in reference to the idea of a generalized notion of limit which, on the one hand, can make sense in more abstract settings than categories (e.g., <a href="http://ncatlab.org/nlab/show/semicategory">Mitchell's semicategories</a>, or even Ehresmann's neocategories [1]), and on the other hand, _may perhaps_ be an effective surrogate of limits in situations where limits do not exist and neither <a href="http://ncatlab.org/nlab/show/weak+limit">weak limits</a> nor sublimits (see note 2) are likely to represent, for some reason, a satisfactory alternative. So, here are my questions:

>> **Q1.** Is $\zeta \ast \bf C$, up to equivalence, a disguised form/a special instance of anything familiar to you? **Q2.** Is it maybe a particular colimit (see note 3)? **Q3.** Does it satisfy any "obvious" universal property?

Thanks in advance, as always, for any hint.

**Remarkably useless notes.** (1) Ordinals have nothing special here, but since I'm working in TG, there is no loss of generality in using ordinals as indexing sets. Also, feel free to think of $\zeta$ as a binary operation on $\alpha$, if you prefer. (2) Sublimits are the same as weak limits, save for the fact that we replace "at least" with "at most". (3) Say, as an object of ${\bf Cat}(\mathcal U)$, for a sufficiently large universe $\mathcal U$.

*Bibliography.*

[1] A. Bastiani and C. Ehresmann, _Categories of sketched structures,_ Cahiers Topologie Géom. Différ. Catégoriques, Vol. 13, No. 2 (1972), 104-214.