I'm completing a paper about (Mitchell's) semicats (well, not exactly, but let's say so for simplicity), and as a motivational example I'd like to mention at some point that the monic/epic morphisms of the semicat of real/complex normed spaces and (linear) compact operators between them (with the obvious source and target maps and the equally obvious composition) are exactly those that one is expected to get. So my question is:
Is there anything in the literature taking the point of view of semicats in the study of compact operators, in such a way that I can cite it (at least for the sake of comparison)?
Feel free to extend the same question to other objects of interest in functional analysis such as real/complex normed spaces and (strictly) contractive linear operators or (topological) pointed spaces and compactly supported base maps. I don't expect anything like Helemskii's Lectures and Exercises on Functional Analysis, but on the other hand I find it a little bit surprising that nobody has already tried to pursue this line of thought, and arguing that the reason for this "gap" may be due to the fact that "semicats are not really more general than cats", since "there exists a functorial way to turn them into a category", is just another instance of the principle of explosion.
Added later.  Loosely speaking, a semicat is a not-necessarily-unital category. For what it is worth, and to the best of my knowledge, the notion was first introduced by B. Mitchell in The dominion of Isbell, TAMS, Vol. 167 (1972), 319-331.  Monic and epic arrows in a semicat are defined in the very same way as monic and epic arrows in categories.  If necessary (though I don't think so): By a compact operator between $\mathcal K$-normed modules, where $\mathcal K = (\mathbb K, |\cdot|)$ is a normed rng (here, just a rng endowed with an absolute value), I mean a triple $f: \mathcal M_1 \to \mathcal M_2$ for which $\mathcal M_i = (\mathbb M_i, \|\cdot\|_i)$ is a normed (left) module over $\mathcal K$ and $f: \mathbb M_1 \to \mathbb M_2$ is a homomorphism of (left) $\mathbb K$-modules such that the image of any bounded subset of $\mathcal M_1$ under $f$ is relatively compact in $\mathcal M_2$.