The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a semigroup $S$, a set $A$, and a mapping of the elements of the semigroup $S$ to functions from the set $A$ to itself. But the analogous result for inverse semigroups requires partial symmetries, i.e. partial functions instead of total functions. But if we allow partial functions, then what do we do with removable singularities?

Here comes the desire to turn all this into "somebody else's problem", by using a general definition of semigroup action in terms of category theory. We can then work in the category of sets and (total) functions, if we don't need partial functions for the semigroup action. If we need partial functions, then we can work in the category of sets and partial functions. And if we worry about removable singularities, then we can work in the appropriate category where these are removable.

But how should such a definition look like? I'm not sure, but let's look at a (potential) similar definition for groupoid action: A groupoid action relative to a category $\mathcal C$ would be a groupoid $\mathcal G$, a mapping $m_O$ of the objects of $\mathcal G$ to objects of $\mathcal C$, and a mapping $m_A$ of the arrows $a:X\to Y$ from the groupoid $\mathcal G$ to arrows $m_A(a):m_O(X)\to m_O(Y)$ of the category $\mathcal{C}$.

One problem I have is that if there were such a thing as a semigroupoid, it would be easy to interpret a small semigroupoid (and hence also a small groupoid) as a semigroup. (Add a new absorbing element and use it to define the result of any undefined composition from the semigroupoid.) But can one define a semigroup action in such a way that also the semigroupoid action can be interpreted appropriately in terms of semigroup action?

One potential issue of the proposed "solution" is that it is unclear whether categories of partial homomorphisms are always as easy to define as in the case of sets. Is there really a "canonical" category of topological spaces and partial continuous maps? Wouldn't it be better to define the concept of partial function from within category theory, for example by marking the "restriction" homomorphism between the objects of a category as an explicitly given (poset) subcategory? That would at least be a "canonical" definition, even if it cannot cope with the issue of removable singularities.