I'm a physicist trying to study Topological Quantum Field Theory (TQFT), so apologies if the following has some basic mistakes or misuse of terminology. When answering please bear in mind that I'm not fluent in category theory, so concrete examples will help.
In Atiyah's formulation, a TQFT is a functor $Z:d\text{Bord}\to\text{Hilb}$. That is, $Z$ assigns:
\begin{align} \text{Closed compact $(d-1)$-manifolds} &\to \text{f.d. Hilbert spaces} \newline \text{$d$-dimensional bordisms between manifolds} &\to \text{Unitary maps between corresponding Hilbert spaces} \end{align}
To implement locality, further structure should be added, leading to the notion of an extended TQFT. An extended TQFT is supposed to assign categories to manifolds of each codimension, possibly with boundaries and corners. To illustrate this, take $d=2$. In Atiyah's TQFT, we can compute things like the following:
In an extended TQFT, we can compute all the above, but in addition we should now be able compute things like: 
I used filled-in and empty points to denote incoming and outgoing. Note that a manifold gets assigned to an object in the category corresponding to its boundary: $Z(M)\in Z(\partial M)$. So for example, Fig 2a is some object in the category $Z(\bullet)\otimes Z(\circ)$. Fig 2c is meant to show a bordism between two line segments, which is an example of a bordism-between-bordisms, and has corners (maybe the standard way to draw this is as a square? I'm not sure if the corners should look pinched like that, but it makes sense to me...).
Now, given some physical theory -- for example Chern-Simons in 3D, or open-closed strings in 2D, or perhaps a 2D gauge theory with action like $\int F$ -- how do we actually calculate the value of diagrams like Fig 2c? Previously I've only ever evaluated $Z$ on manifolds without corners...
I'd be satisfied with an answer that gives any concrete example of a TQFT where we know how to evaluate Fig 2c. Even better would be some procedure for starting with a theory (e.g. Chern-Simons) and working out what $Z$ should assign to manifolds with corners.
Thanks!