Being "affine" in this case does not make much sense,
because the hyperkaehler deformation is a complex manifold, without
a fixed algebraic structure. Simpson produced an example of a
hyperkaehler deformation of a space of flat bundles
admitting several algebraic structures, both
inducing the same Stein complex structure; one of them
is affine, another has no global algebraic functions.
In fact, the space F of flat line bundles on elliptic curve
(with an appropriate algebraic structure, defined by
Simpson) is an example of such a manifold,
it is biholomorphic to $C^*\times C^*$, but
this biholomorphic equivalence is not algebraic,
and F has no global algebraic functions.
However, you can show that a hyperkaehler deformation
of a resolution of something affine has no non-trivial complex
subvarieties (arXiv:math/0312520), except, possibly, some
hyperkaehler subvarieties The latter don't exist, because
the holomorphic symplectic form $\Omega$ on such a manifold
is is lifted from the base, which is affine, hence $\Omega$
vanishes on all complex subvarieties.
Therefore, a typical fiber of such a deformation is Stein.
Indeed, a hyperkaehler deformation of a
resolution of something affine remains holomorphically convex. To see this
if you produce a function which is strictly plurisubharmonic outside of
a compact set (we have such a function, because we started from something
affine), and apply the Remmert reduction.
I can explain the pictures I usually draw to think of $A_\infty$ functors,
but I don't know if they're standard. Anyway, I'll describe what is
just a rubric for ingesting the long formulas, nothing more.
Let's consider first the Yoneda embedding $Y$, which re-thinks an object $L$
in an $A_\infty$-category $A$ as an $A$-module, or functor from $A^{op}$ to
chain complexes. So $Y_L(M) = hom_A(M,L).$
I confess that when I confront these formulas/concepts, I always think
in terms of the Fukaya category, which is very amenable to pictures
and for which the $A_\infty$ structures are geometric.
So I draw a curve on a piece of paper and label it $L$. (The curve
is literally a Lagrangian submanifold of my ${\mathbb R}^2$ piece of
paper.) When I want
to think of $L$ in terms of its Yoneda image, I draw the SAME curve, but as a
squiggly line.
So what is the data that the squiggly line gives us? For each object $M$
(a regular curve on my paper), we have the intersection points, which form a
graded vector space $hom_A^*(M,L).$ This vector space has the
structure of a chain complex (Floer), with differential given by
football-shaped bi-gons with one regular side and one squiggly side.
For a pair of other objects, $M_1, M_2,$ we get a map
$$\mu^2: hom_A(M_2,L)\otimes hom_A(M_1,M_2) \rightarrow hom_A(M_1,L),$$
and so on for all the structure of a module (section 1j, p. 19).
For the Fukaya category,
the equations 1.19 follow (for non-squiggly lines) from studying degenerations
of 1-parameter families of holomorphic polygons. Now squigglifying those same pictures
gives 1.19 for an arbitrary module, and the equations are similar for
not just modules but arbitrary functor between two $A_\infty$-categories.
What data do we have if we have two squiggly lines $L_1$ and $L_2$?
They should intersect at a morphism between functors (and it should have a degree).
This morphism of functores gives more data, using the Fukaya perspective.
If we added one normal line $M$, we'd have the spaces $Y_{L_1}(M)$ and $Y_{_2}(M)$,
and have a triangle which is a map between them. Higher polygons and the
relations between them (by considering one-parameter families) should
give you all the equations and give you a hint as to verify them.
(But no promises!)
Hope that lengthy and pretty vague description was worth our time.
(Oh, geez, this was a March 11 question? Probably stale by now!)
Best Answer
Your structure can be described as a "category without identity", which has been given the names "semicategory" and "semigroupoid" presumably due to independent discoveries.
Some Googling suggests the term "semicategory" came first, in a 1972 TAMS paper by Mitchell. The name is motivated by applying an analogy connecting groups and semigroups to categories (as categories without identities or inverses), and it seems to be popular among people who study categories.
The term "semigroupoid" seems to have appeared first in Tilson, Categories as Algebra: an essential ingredient in the theory of monoids in Journal of Pure and Applied Algebra 48 (1987) 83-198. The name is motivated by applying an analogy connecting groups and groupoids to semigroups (as semigroups with multiple objects), and it seems to be popular among people who study semigroups.
I think the analogy is a bit weak on the semicategory side, since categories don't straightforwardly generalize groups. I'm not in charge, though.
John Baez points out in his TWF week 296 that there is a canonical way to make a category out of a semicategory, by formally adding, for each object, an identity element to the set of morphisms from that object to itself (and preserving all other morphism sets and composition laws). Any previously existing identities become idempotents. He notes that the categories formed this way are distinguished among all categories by the property that all invertible morphisms are identities. In particular, this process of formally adding identities is reversible in a canonical way, and no hell will break loose.