*I was wondering whether the following category already has been used somewhere and whether it already has been named*.


Let us fix a field $k$ (or more generally a ring). An object is just a $k$-vector space and a morphism $f:U\rightarrow V$ consists of a choice of a sub-vectorspace $V\_f\subset V$ and a well-defined linear map $\overline{f}:U\rightarrow V/V\_f$.


The composition of $f:U\rightarrow V$ and $V\rightarrow W$ is given by
$$W\_{g \circ f} = p\_{W\rightarrow W/W\_g}^{-1}(g(V\_f)),$$
$$\overline{g\circ f}(x) := [\overline{g}(y)],$$
where $p\_{W\rightarrow W/W\_g}$ denotes the canonical projection $y\in V$ is a representative of $\overline{f}(x)\in V/V\_f$.


The reason behind this is that I wanted to compose maps this way quite often, and one could either


* give a new name to $V/V\_f\rightarrow (U/W\_g)/g(V\_f)$ and work in the category of vectorspaces. Now I would have to introduce a lot of new names for the map $g$ depending on with which maps I compose them, which makes things worse to read.
* Try to lift each map $U\rightarrow V\rightarrow V/V\_f$ to a map $U\rightarrow V$. This involves a choice which might affect the rest of the arguments. Now one has to argue that things do not depend on the choice for no other reason than that we have introduced
* The naive idea that there should be a functor from this category back to vectorspaces sending $f$ to $\overline{f}$ does not work, since this does not respect composability (The target of $f$ is $V/V\_f$ and the source of $g$ is $V$.


*So I think using this category might be the cleanest way and so I am wondering whether it has already been used/named before.*



category of non-welldefined linear maps
As Denis Nardin pointed out in the comments, the name *total correspondence* is a good choice.


