A new category $C^*$ from a given category $C$ For a given category $C$ define the category $C^*$ as follows: the objects of $C^*$ are those of $C$; for given objects $u,v$, the $C^*$-morphisms $u\to v$ are all finite sequences $(a_1,\dots,a_n)$ of morphisms $a_i\in Mor(C)$ such that $a_1:u\to x$ and $a_n:y\to v$ where $x$ and $y$ are arbitrary objects (no other restrictions). The composition of two composable morphisms is $$(a_1,\dots,a_m)(b_1,\dots,b_n)=(a_1,\dots,a_mb_1,\dots,b_n)$$ (where $a_mb_1$ are composed in $C$). My question: is there an established name for the category $C^*$?

Update:

1) Motivation: the construction $C\mapsto C^*$ appears as a tool of proof in my research and I wanted to know if, where and for which purpose this construction appears in literature, in order to insert some references. Intuitively I would say this is somehow the free category generated by $C \cup (C\times C)$ modulo the relations which hold in $C$.

2) Intuitive background: I have algebraic terms of a certain type which can be interpreted as instructions what to do in a certain category $C$. There are two sorts of instructions:

a) type $w$: they tell me I should compose certain morphisms of $C$  the result of which  is the morphism $a_1$, say, and thereby I run through the underlying graph $\Gamma$ of $C$

b) type $w^{\mathfrak m}$: they say I should jump elsewhere in the graph $\Gamma$ 

The application of such instructions alternatingly ends up with a tuple $(a_1,a_2,\dots,a_n)$ as in the question. The category $C^*$ seems to model exactly this behaviour. For technical reasons, I allow in type b) "empty jumps", that is, even if two consecutive morphisms $a_i$ and $a_{i+1}$ are composable in $C$ I would like to distinguish between $\dots a_i,a_{i+1}\dots$ and $\dots a_ia_{i+1}\dots$.
It looks to me like the two categories are exactly same! The objects are the same so take 2 arbitrary objects A and B, then you can see that the Hom(A,B) in C and C* are in bijection.
It looks to me like the two categories are exactly same! The objects are the same so take 2 arbitrary objects A and B, then you can see that the Hom(A,B) in C and C* are in bijection.