The following was given as an example for a semigroup without an identity:
Finite sets of matrices of varying dimensions, where the product $A*B=\{PQ \mid P \in A, Q \in B \text{ and } dim(Q)=codim(P)\}$, where $dim$ and $codim$ are the dimensions of the source and the target spaces of a matrix.
Editor clarification: $dim(Q)$ is actually the dimension of the target space of the matrix $Q$, $codim(P)$ is the dimension of the source space of the matrix $P$ in the citation.
It took me some seconds before I understood what is meant by that example. And then I thought that this construction could be applied to any small category, not just matrices.
So for a small category $C$, let $S_C:=\{M\subset\operatorname{hom}(C):|M|<\infty \}$ and for $A,B\in S_C$ define $A*B:=\{f\circ g:f\in A, g\in B,\operatorname{source}(f)=\operatorname{target}(g)\}$. Then $S_C$ together with the operation $*$ is a semigroup. (The identity would be $E=\{\operatorname{id}_A:A\in\operatorname{ob}(C)\}$. If the category $C$ has only a finite number of objects, we have $E\in S_C$.)
What I find even more interesting is that $S_c:=\{A\in S_C:|A|\leq1\}$ is a sub-semigroup of $S_C$. Because $S_c=\{\varnothing\}\cup\{\{f\}:f\in\operatorname{hom}(C)\}$, we have a "natural" correspondence between the semigroup $S_c$ and $\operatorname{hom}(C)\cup\{0\}$. Here $0\notin\operatorname{hom}(C)$ is an absorbing element corresponding to the empty set $\varnothing\in S_c$. The semigroup operation of $S_c$ is just the composition of the corresponding morphisms if this is defined, or the empty set otherwise.
Given that this construction was extracted from an answer to a question about interesting semigroups, it can't be totally unknown. However, I would like to know whether this construction has a special name, where it is described, and whether it has "interesting" applications.