Let ${\bf C}$ be a category, $\alpha$ an ordinal number, and $\zeta$ a 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., Mitchell's semicategories, 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 weak limits 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.